An extensive photometric study of the Blazhko RR Lyrae star DM Cyg★

Abstract

ABSTRACTDM Cyg, a fundamental mode RRab star, was observed in the 2007 and 2008 seasons in the framework of the Konkoly Blazhko Survey. Very small amplitude light-curve modulation was detected with 10.57 d modulation period. The maximum brightness and phase variations do not exceed 0.07 mag and 7 min, respectively. In spite of the very small amplitude of the modulation, beside the frequency triplets characterizing the Fourier spectrum of the light curve, two quintuplet components were also identified. The accuracy and the good phase coverage of our observations made it possible to analyse the light curves at different phases of the modulation separately. Utilizing the inverse photometric Baade–Wesselink method, we could detect very small systematic changes in the global mean physical parameters of DM Cyg during its Blazhko cycle. The detected changes are similar to what we have already found for a large modulation amplitude Blazhko variable, MW Lyrae. The amplitudes of the detected changes in the physical parameters of DM Cyg are only about 10 per cent of those that have been found in MW Lyr. This is in accordance with its small modulation amplitude being about one-tenth of the modulation amplitude of MW Lyr.The pulsation period of DM Cyg has been increasing by a rate of β= 0.091 d Myr−1 during the hundred-year time base of the observations. Konkoly archive photographic observations indicate that when the pulsation period of the variable was shorter by Δppuls= 5 × 10−6 d the modulation period was longer by Δpmod= 0.066 d than today.

Full text

Page 1

arXiv:0904.4129v1 [astro-ph.SR] 27 Apr 2009
Mon. Not. R. Astron. Soc. 000, 1–12 (2009)Printed 27 April 2009(MN LATEX style file v2.2)
An extensive photometric study of the Blazhko RR Lyrae
star DM Cyg⋆
J. Jurcsik1Zs. Hurta1,2,´A. S´ odor1, B. Szeidl1, I. Nagy2, K. Posztob´ anyi4,
M. V´ aradi1,3, K. Vida1,2, B. Belucz2, I. D´ ek´ any1, G. Hajdu2, Zs. K˝ ov´ ari1, E. Kun5
1Konkoly Observatory of the Hungarian Academy of Sciences, H-1525 Budapest PO Box 67, Hungary
2E¨ otv¨ os University, Dept. of Astronomy, H-1518 Budapest PO Box 49, Hungary
3Observatoire de Geneve, Universite de Gen` eve, CH-1290, Sauverny, Switzerland
4AEKI, KFKI Atomic Energy Research Institute, Thermohydraulic Department, H-1525 Budapest 114, PO Box 49, Hungary
5Department of Experimental Physics and Astronomical Observatory, University of Szeged, 6720 Szeged, D´ om t´ er 9, Hungary
Accepted 2009 December 15. Received 2009 December 14; in original form 2009 January 22
ABSTRACT
DM Cyg, a fundamental mode RRab star was observed in the 2007 and 2008
seasons in the frame of the Konkoly Blazhko Survey. Very small amplitude light curve
modulation was detected with 10.57 d modulation period. The maximum brightness
and phase variations do not exceed 0.07 mag and 7 min, respectively. In spite of the
very small amplitude of the modulation, beside the frequency triplets characterizing
the Fourier spectrum of the light curve two quintuplet components were also identified.
The accuracy and the good phase coverage of our observations made it possible to
analyse the light curves at different phases of the modulation separately. Utilizing the
IP method (S´ odor, Jurcsik and Szeidl, 2009) we could detect very small systematic
changes in the global mean physical parameters of DM Cyg during its Blazhko cycle.
The detected changes are similar to what we have already found for a large modulation
amplitude Blazhko variable MW Lyrae. The amplitudes of the detected changes in the
physical parameters of DM Cyg are only about 10% of that what have been found in
MW Lyr. This is in accordance with its small modulation amplitude being about one
tenth of the modulation amplitude of MW Lyr.
The pulsation period of DM Cyg has been increasing by a rate of β = 0.091dMyr−1
during the hundred-year time base of the observations. Konkoly archive photographic
observations indicate that when the pulsation period of the variable was shorter by
∆ppuls= 5·10−6d the modulation period was longer by ∆pmod= 0.066 d than today.
Key words:
DM Cyg – techniques: photometric – methods: data analysis –
stars: variables: other – stars: horizontal branch – stars: individual:
1 INTRODUCTION
Utilizing our full access to an automatic 60 cm telescope we
have obtained extended multicolour observations of many
fundamental mode RR Lyrae variables showing light curve
modulation (the Blazhko effect) during the past five years.
Detailed analyses of some of our targets were already pub-
lished in Jurcsik et al. (2005, 2006, 2008, 2009). Our obser-
vations are the first multicolour photometric data which are
accurate and dense enough to allow not only to determine
⋆Based on observations collected with the automatic 60 cm tele-
scope of Konkoly Observatory, Budapest, Sv´ abhegy
the frequencies appearing in the Fourier spectra of the light
curves, but also to find changes in the global mean physical
parameters of the stars (L,Teff,R) in different phases of
the modulation. In order to extract changes in the phys-
ical parameters exclusively from multicolour photometric
data we developed an inverse photometric Baade-Wesselink
method (IPM; S´ odor, Jurcsik & Szeidl 2009). The IPM was
successfully applied to determine changes in the mean global
physical parameters of MW Lyr during its Blazhko cycle in
Jurcsik et al. (2009).
DM Cyg, a short period (P = 0.42 d) RRab star was
selected to be observed in the frame of the Konkoly Blazhko
Survey because it had been announced to show phase modu-

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2
J. Jurcsik et al.
Table 1. Konkoly photometric observations of DM Cyg.
HJD
2400000+
mag∗
detectorfilter
54265.45479
54265.46045
54265.46611
...
−0.249
−0.404
−0.501
...
CCD
CCD
CCD
...
V
V
V
...
∗Relative magnitudes for the CCD and photoelectric data.
Table 2. Normal maximum timings derived from the Konkoly
photographic, photoelectric and CCD observations.
JDNormal maximum timingsobs.
2427667 − 2429226
2434186 − 2436373
2443747 − 2444133
2454265 − 2454407
2454597 − 2454711
2429078.482
2435342.347
2443777.333
2454334.3467
2454661.4210
pg
pg
pe
CCD
CCD
lation with a period of 26 d by Lysova & Firmanyuk (1980),
but neither the NSVS data (Wo´ zniak et al. 2004) nor the
complete data set of its maximum timings show phase mod-
ulation with this period (S´ odor & Jurcsik 2005).
Though DM Cyg is a relatively bright (V = 10−11 mag)
RR Lyrae star, complete, accurate, multicolour light curve
of its pulsation has never been published. Its period change
was, however, regularly monitored by different groups of ob-
servers. The GEOS database1lists 260 maximum times of
DM Cyg between 1900 and 2008.
In the present paper we publish our extended CCD ob-
servations of DM Cyg and the results of the analysis of its
light curve modulation. Archive photographic and photo-
electric Konkoly data are also processed.
2 DATA
CCD observations were obtained with the automated 60
cm telescope of the Konkoly Observatory, Sv´ abhegy, Bu-
dapest equipped with a Wright Instruments 750 × 1100
CCD camera and BV IC filters. Measurements were taken
on 81 nights between July 2007 and Sept 2008. About
3100 data points in each band were gathered. Exposition
times were 200, 60 and 40 sec or a bit longer depend-
ing on the sky transparency in the BV IC bands, respec-
tively. Data reduction was performed using standard IRAF2
packages. Aperture photometry of DM Cyg (21:21:11.548
+32:11:28.71) and several neighbouring stars were carried
out in order to check the stability of the photometry and
the constancy of the comparison stars. The relative mag-
nitudes of DM Cyg measured to the mean magnitudes of
C1=GSC2.2 N0330220980 (21:21:30.731 +32:13.05.78), and
1http://dbrr.ast.obs-mip.fr/maxRR.html
2IRAF is distributed by the National Optical Astronomy Obser-
vatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation.
-1
-0.75
-0.5
-0.25
0
0.25
54300 54400 54500 54600 54700
delta V
JD-2400000
A
-1
-0.75
-0.5
-0.25
0
0.25
-0.5 -0.25 0 0.25 0.5
deltaV
pulsation phase
B
-1
-0.75
-0.5
-0.25
0
0.25
-0.5 -0.25 0 0.25 0.5
deltaV
modulation phase
C
Figure 1. Delta V magnitudes versus Julian Date, data phased
with the 0.419863d pulsation and the 10.57d modulation periods
are shown in panels A, B and C, respectively.
C2=GSC2.2 N03302207371 (21:20:59.483 +32:13.01.63) are
used in the analysis. The Tycho B and V magnitudes of the
comparison stars C1 and C2 are BT = 12.705, VT = 11.799,
and BT = 13.081,VT = 12.150, respectively (Hog et al.
2000). Second order extinction correction of the data were
applied in the B band. Relative magnitudes are transformed
to standard BV IC magnitudes. More details on the reduc-
tion procedure are given in Jurcsik et al. (2008).
Archive photoelectric and photographic data obtained
with the 60 cm telescope in 1978 and with a 16 inch astro-
graph between 1934 and 1958 are also utilized. Photoelectric
observations were obtained on 4 nights, the BV magnitudes
were measured relative to GSC2.2 N03302207371. The pho-
tographic measurements comprise data from 40 nights, the
plates were evaluated using GSC2.2 B magnitudes of the
surrounding stars.
Table 1 shows a sample of the Konkoly photometric
data of DM Cyg. All the photometric observations are avail-
able in the online version of the journal as Supplementary
Material.
Seasonal normal maximum timings from each data set
have been determined for five epochs, these data are listed
in Table 2.
3RESULTS
The accurate CCD observations reveal that the light curve of
DM Cyg is not stable. It shows small amplitude modulation,
the variations in maximum brightness and phase are only
about 0.07 mag and 0.005 d (7 min, 0.012 pulsation phase),
respectively.
The V light curve of DM Cyg and the results obtained
folding the data with the pulsation and modulation periods
are shown in the three panels of Fig. 1. The phased resid-
ual light curve after the removal of the mean pulsation light
variation is given in Fig. 2. This figure shows the differences

Page 3

Photometric study of the Blazhko star DM Cyg
3
-0.9-0.9
-0.85-0.85
-0.5-0.5-0.25-0.25 0 0 0.25 0.25 0.5 0.5
maximum brightness
Blazhko phaseBlazhko phase
maximum brightness
-0.01 -0.01
-0.005 -0.005
0 0
0.005 0.005
0.01 0.01
-0.5 -0.5-0.25-0.25 0 0 0.25 0.25 0.5 0.5
maximum phase
Blazhko phaseBlazhko phase
maximum phase
-0.9-0.9
-0.85-0.85
-0.01-0.01 -0.005-0.005 0 0 0.005 0.005 0.01 0.01
maximum brightness
maximum phasemaximum phase
maximum brightness
Figure 3. Maximum V brightness and maximum phase values phased according to the Blazhko period are plotted in the top-left and
bottom-left panels, respectively. Their typical errors are 0.005 mag and 0.002-0.004 phase. The right-hand panel shows the maximum
brightness vs maximum phase data. The progression of data during the Blazhko cycle in this diagram is anti-clockwise. Dashed lines
connect the mean values in 10 phase bins of the modulation.
-0.08-0.08-0.08
-0.06-0.06-0.06
-0.04-0.04-0.04
-0.02-0.02-0.02
0 0 0
0.02 0.02 0.02
0.04 0.04 0.04
0.06 0.06 0.06
0.08 0.08 0.08
-0.5-0.5 -0.5-0.25-0.25-0.25 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5
Vresidual
pulsation phasepulsation phasepulsation phase
Vresidual
Vresidual
-0.5-0.5-0.5-0.25-0.25-0.25 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5
Vresidual
pulsation phasepulsation phasepulsation phase
Vresidual
Vresidual
Figure 2. Residual V light curve of DM Cyg after removing the
pulsation components from the data according to the light curve
solution given in Table 3. Residuals of the smallest (stars) and
the largest (diamonds) amplitude phases are shown with large
symbols in the left-hand and right-hand panels, respectively. Note
that the residuals have larger amplitude at around the middle of
the rising branch of the pulsation (phase = −0.1) than around
pulsation maxima (phase = 0) due to phase modulation.
of the amplitude of the modulation at different phases of the
pulsation. The well observable amplitude of the modulation
is concentrated to a narrow phase range of the pulsation
around minimum, rising branch and maximum. A similar
behaviour has been already found in RR Gem and SS Cnc,
in two other small modulation amplitude Blazhko variables
(Jurcsik et al. 2005, 2006). The amplitude of the the resid-
ual variation is the largest at around the middle of the rising
branch of the pulsation light curve (phase = −0.1) indicat-
ing phase modulation of the rising branch as well. There is a
difference between the phase of the amplitude modulation of
the maximum of the light curve and the phase of the phase
modulation of the same feature. In particular the maximum
positive displacement of the timings of the maximum bright-
ness precedes the occurrence of the brightest maximum by
about 0.25 Blazhko phase as it is shown Fig. 3.
The elements of the pulsation and the modulation are:
Tmax puls= 2454312.514 [HJD] + 0.419863 · Epuls,(1)
and
Tmax Bl= 2454312.514 [HJD] + 10.57 · EBl.(2)
Epulsand EBldenote the epoch number of the pulsation and
modulation cycles, respectively.
The pulsation and modulation periods correspond to
the mean values of the frequencies of the light curve so-
lutions of the B,V and IC data using ‘locked’ frequency
solutions of the kf0 pulsation and kf0± fm, and fm modu-
lation frequency components (i.e., the modulation side lobe
components are at the positions of the linear combination
frequencies).
Data analysis was performed using the different appli-
cations of the MUFRAN package (Koll´ ath 1990), a linear
combination fitting program developed by´A. S´ odor, and
the linear and nonlinear curve fitting abilities of gnuplot3.
3http://www.gnuplot.info/

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4
J. Jurcsik et al.
0.000
0.200
0.400
0.600
0.800
1.000
-25-20-15 -10-5 0 5 10 15 20 25
Spectral Window
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0 5 10 15 20 25 30 35 40 45 50
amplitude
0.000
0.002
0.004
0.006
0.008
0.010
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency
0.0000
0.0016
0.0004
0.0008
0.0012
0.0016
k = 10
0.0000
0.0016
0.0004
0.0008
0.0012
k = 11
0.0000
0.0004
0.0008
0.0012
kf0 - 2fm
kf0 - fm
kf0
kf0 + fm
kf0 + 2fm
k = 12
frequency
0.0000
0.0016
0.0004
0.0008
0.0012
0.0016
k = 25
0.0000
0.0016
0.0004
0.0008
0.0012
k = 26
0.0000
0.0004
0.0008
0.0012
kf0 - 2fm
kf0 - fm
kf0
kf0 + fm
kf0 + 2fm
k = 27
frequency
Figure 4. The left-hand panels show the spectral window, the amplitude spectrum and the residual spectrum after the removal of the
pulsation components for the CCD V data. The middle and right-hand panels show residual spectra in the vicinity of the 10f0,11f0,
and 12f0 (k = 10,11,12) and of the 25f0,26f0, and 27f0 (k = 25,26,27) frequencies, respectively. The spectrum has been prewhitened
for the pulsation components up to the 24th order, and in the right hand panels for the detected modulation components, too. These
spectra indicate that modulation frequency components are present only in the first 10-11 orders, while the pulsation can be accurately
described if harmonic frequency components are taken into account up to as high as the 27th order. For clarity, vertical grids denote the
positions of the kf0,kf0+ fm and kf0− fm frequencies.
0.000
0.001
k = 1
B
0.000
0.001
kf0 - 2fmkf0 - fm
kf0
kf0 + fmkf0 + 2fm
k = 2
frequency
0.000
0.001
k = 1
V
0.000
0.001
kf0 - 2fmkf0 - fm
kf0
kf0 + fmkf0 + 2fm
k = 2
frequency
Figure 5. Fourier spectra of the V and B residual data in
the vicinity of f0 and 2f0. The pulsation and the kf0 ± fm
modulation frequencies have been removed. The f0− 2fm and
2f0− 2fm quintuplet components appear in these residual spec-
tra with ∼ 0.001 mag amplitude.
3.1 The light curve solution
The Fourier amplitudes and phases of the pulsation and
modulation frequency components identified in the spectra
of the BV IC light curves of DM Cyg are summarized in
Table 3. The Fourier decompositions use sin terms and the
initial epoch corresponds to one of the brightest maxima of
the modulated light curve, T0 = 2454312.514. The errors of
the amplitudes are ∼ 0.0002 mag, the errors of the phases for
the time transformed V data (see the details later) are given
in the last column. The pulsation components are detectable
up to the 27th order, while the modulation side frequencies
(kf0±fm) diminish at around the 11th order as documented
in Fig. 4. The modulation frequency fm is present without
question in each of the B,V,I spectra with similar ampli-
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 5 10 15
Amplitude ratio
harmonic order (k)
Akf0/Af0
Akf0+fm/Af0+fm
Akf0-fm/Af0-fm
Figure 6. Amplitude decrease of the pulsation and modulation
side lobe frequencies at different orders shown as amplitudes nor-
malised to the amplitudes of the first order (k=1) components.
The decrease of the amplitudes of the harmonic components of
the pulsation is exponential like, while the decrease of the ampli-
tudes of the side lobe frequencies is more linear, with irregular
character in the low orders.
tudes as the 4th order negative and the 9th order positive
modulation side lobe components have. Quintuplet frequen-
cies are detected at f0 − 2fm and 2f0 − 2fm as shown in
Fig. 5.
The 51 frequencies listed in Table 3 fit the data with 6-8
mmag residual scatter in each band, which is about the level

Page 5

Photometric study of the Blazhko star DM Cyg
5
Table 3. Fourier amplitudes and phases of the pulsation and modulation frequencies in DM Cyg. Initial epoch corresponds to maximum
of both the pulsation and the modulation light variations as given in Eqs. 1 and 2.
frequencyBVIc
V∗
tt
σϕ(Vtt)
ampϕ ampϕampϕampϕ
fm
f0− 2fm
f0− fm
f0
f0+ fm
2f0− 2fm
2f0− fm
2f0
2f0+ fm
3f0− fm
3f0
3f0+ fm
4f0− fm
4f0
4f0+ fm
5f0− fm
5f0
5f0+ fm
6f0− fm
6f0
6f0+ fm
7f0− fm
7f0
7f0+ fm
8f0− fm
8f0
8f0+ fm
9f0− fm
9f0
9f0+ fm
10f0− fm
10f0
10f0+ fm
11f0
11f0+ fm
12f0
13f0
14f0
15f0
16f0
17f0
18f0
19f0
20f0
21f0
22f0
23f0
24f0
25f0
26f0
27f0
0.094607
2.192515
2.287122
2.381729
2.476337
4.574244
4.668851
4.763459
4.858066
7.050581
7.145188
7.239795
9.432310
9.526917
9.621525
11.814039
11.908647
12.003254
14.195768
14.290376
14.384983
16.577498
16.672105
16.766713
18.959227
19.053834
19.148442
21.340956
21.435564
21.530171
23.722686
23.817293
23.911900
26.199022
26.293630
28.580752
30.962481
33.344210
35.725940
38.107669
40.489398
42.871127
45.252857
47.634586
50.016315
52.398045
54.779774
57.161503
59.543233
61.924962
64.306691
0.0026
0.0015
0.0036
0.4903
0.0132
0.0014
0.0035
0.2598
0.0105
0.0050
0.1545
0.0100
0.0024
0.0816
0.0086
0.0019
0.0510
0.0069
0.0012
0.0338
0.0056
0.0015
0.0185
0.0040
0.0003
0.0114
0.0030
0.0009
0.0048
0.0021
0.0004
0.0016
0.0015
0.0006
0.0009
0.0019
0.0024
0.0026
0.0023
0.0024
0.0021
0.0019
0.0016
0.0018
0.0009
0.0014
0.0009
0.0009
0.0006
0.0010
0.0007
1.678
3.363
2.694
3.944
3.822
0.664
2.355
4.109
3.511
3.173
4.475
3.928
3.179
4.970
4.464
3.800
5.142
4.868
4.312
5.592
5.159
5.147
5.861
5.601
0.174
6.253
5.913
0.444
0.309
0.204
1.793
0.291
0.576
4.556
1.438
4.559
4.971
5.317
5.712
6.073
6.242
0.187
0.452
0.765
0.818
1.161
1.432
2.064
2.156
2.759
2.871
0.0017
0.0008
0.0022
0.3532
0.0096
0.0008
0.0026
0.1919
0.0080
0.0034
0.1160
0.0079
0.0019
0.0628
0.0063
0.0016
0.0391
0.0053
0.0011
0.0255
0.0045
0.0010
0.0146
0.0034
0.0006
0.0089
0.0026
0.0006
0.0037
0.0019
0.0005
0.0012
0.0013
0.0009
0.0010
0.0014
0.0021
0.0024
0.0023
0.0022
0.0019
0.0017
0.0014
0.0014
0.0012
0.0010
0.0010
0.0010
0.0008
0.0005
0.0006
1.637
2.989
2.618
3.874
3.854
0.275
2.404
4.090
3.572
3.187
4.476
3.988
3.237
4.971
4.437
3.799
5.171
4.889
4.130
5.617
5.169
4.585
5.921
5.591
5.416
0.030
5.954
0.525
0.482
0.034
1.018
0.760
0.624
4.183
0.835
4.526
4.891
5.239
5.560
5.883
6.154
0.166
0.546
0.920
1.023
1.394
1.644
2.306
2.556
2.907
2.946
0.0014
0.0005
0.0011
0.2170
0.0061
0.0004
0.0012
0.1189
0.0049
0.0024
0.0728
0.0048
0.0012
0.0396
0.0042
0.0010
0.0247
0.0035
0.0008
0.0162
0.0027
0.0010
0.0088
0.0020
0.0004
0.0050
0.0016
0.0009
0.0024
0.0013
0.0008
0.0007
0.0009
0.0009
0.0008
0.0015
0.0017
0.0019
0.0016
0.0020
0.0015
0.0015
0.0015
0.0011
0.0011
0.0010
0.0007
0.0006
0.0007
0.0004
0.0007
1.517
3.939
2.839
3.670
3.975
0.976
2.513
4.029
3.619
3.270
4.466
4.003
3.230
4.972
4.466
3.646
5.204
4.862
4.502
5.665
5.102
4.748
5.979
5.667
6.238
0.167
6.054
0.135
0.769
0.248
0.687
2.221
0.652
3.499
1.374
4.265
4.564
5.240
5.429
5.677
6.173
0.062
0.444
0.755
1.094
1.384
1.658
2.073
2.521
2.592
3.139
0.0017
0.0008
0.0048
0.3532
0.0057
0.0008
0.0043
0.1920
0.0038
0.0053
0.1162
0.0040
0.0027
0.0629
0.0036
0.0027
0.0392
0.0030
0.0019
0.0257
0.0028
0.0015
0.0147
0.0022
0.0012
0.0089
0.0018
0.0010
0.0038
0.0015
0.0006
0.0012
0.0011
0.0009
0.0011
0.0015
0.0021
0.0024
0.0024
0.0022
0.0019
0.0017
0.0015
0.0014
0.0013
0.0011
0.0010
0.0010
0.0008
0.0004
0.0006
1.644
2.697
3.675
3.874
4.043
0.573
3.757
4.090
3.284
4.058
4.476
3.772
4.562
4.970
4.222
4.826
5.173
4.874
5.325
5.617
5.047
5.503
5.925
5.527
6.113
0.033
5.879
0.647
0.480
6.239
1.077
0.780
0.573
4.147
0.818
4.462
4.878
5.185
5.521
5.845
6.104
0.106
0.533
0.881
0.947
1.321
1.673
2.259
2.568
2.883
2.961
0.107
0.211
0.037
0.001
0.032
0.219
0.042
0.001
0.047
0.034
0.002
0.044
0.065
0.003
0.050
0.067
0.004
0.060
0.094
0.007
0.064
0.117
0.012
0.080
0.143
0.018
0.099
0.172
0.046
0.113
0.289
0.141
0.162
0.192
0.162
0.116
0.082
0.073
0.073
0.079
0.088
0.096
0.111
0.121
0.132
0.162
0.177
0.177
0.226
0.402
0.258
∗Fourier parameters of the time transformed V data
of the observational noise. No further periodic or stochastic
variation in the residuals is detected.
The decrease of the amplitudes of the pulsation compo-
nents with increasing order is smooth and exponential-like.
The amplitude decrease of the modulation components is
however, different. The amplitudes of the low order mod-
ulation components behave irregularly, while they show a
linear decrease at higher orders (see Fig. 6). Similar be-
haviour of the amplitude decrease of the detected frequen-
cies were demonstrated for RR Gem, SS Cnc and SS For
(Jurcsik et al. 2005, 2006; Kolenberg et al. 2008).
Fig. 7 shows the V light curves belonging to 10 different

Page 6

6
J. Jurcsik et al.
-0.8
-0.6
-0.4
-0.2
0
0.2
∆V [mag]
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.5-0.2500.25
∆V [mag]
-0.5-0.2500.25-0.5-0.25
pulsation phase
00.25-0.5-0.2500.25-0.5-0.2500.250.5
Figure 7. ∆V light curves in 10 phase bins of the modulation. All the observations are also shown with gray colour in each plot. Empty
circles denote artificial data which are used to stabilize the Fourier fits if there is some gap in the observations.
phase bins of the modulation. The first five Fourier param-
eters of the 15th order harmonic fits to these individual V
light curves are plotted in Fig. 8. Comparing the variations
of the Fourier parameters of the light curves of DM Cyg
with that of MW Lyr (Fig. 12 in Jurcsik et al. 2008) the
following differences are conspicuous. The amplitudes of the
detected changes in the Fourier parameters are about one
order of magnitude smaller than the amplitudes of the light
curve variations in MW Lyr. The amplitudes of the varia-
tion of the amplitudes of the f0,...5f0 components decrease
more drastically with increasing order in MW Lyr than in
DM Cyg. This is most probably connected to that the pul-
sation components are detected only up to the 12th order in
MW Lyr, while they can be observed up to the 27th order in
DM Cyg. Nevertheless the phase relations between the am-
plitude modulation and the variation in the phase of the f0
pulsation component are the same for the two stars, the de-
tected changes in the epoch independent phase differences
[ϕ(fk1)] are, however, significantly different. While in DM
Cyg the ϕ(fk1) phase differences show sinusoidal variations
with 0.25 (π/2) phase shift and with larger amplitude than
ϕ(f0), in MW Lyr the amplitudes of the variations of the
ϕ(fk1) components are smaller than the amplitude of ϕ(f0)
and they do not show strictly regular behaviour with the
modulation period.
In Jurcsik et al. (2008) we have introduced a new
method of analysing Blazhko variables’ light curves. It was
shown that if the times of the observations are corrected
according to an appropriate time transformation defined by
the variation of the phase of f0 during the Blazhko cycle,
then the modulation can be separated into phase and am-
plitude modulation components. The time transformed data
showed pure amplitude modulation for MW Lyr. Though the
modulation amplitude of DM Cyg is very small, we have
checked how a similar time transformation influences its
modulation properties. Transforming the times of the ob-
servations according to a continuous harmonic function of
the ϕ(f0) data, the V observations folded with the pulsation
period is compared to the folded light curve of the original
data of DM Cyg in Fig. 9. Enlarged plots of the middle of
the rising branch are inserted in the figures. Though the dif-
ference between the original and the time transformed data
is small, the reduction of the phase modulation component
in the time transformed data is indicated by the narrowness
of the rising branch in the right-hand panel of Fig. 9.
In the last three columns of Table 3 the Fourier ampli-
tudes and phases and the errors of the phases are listed for
the time transformed V data. The amplitudes of the mod-
ulation side frequency components kf0+ fm and kf0− fm
are about the same and the phases of the triplets in the dif-
ferent orders show definite phase coherency according to the
Fourier solution of the time transformed data (see Fig. 10).
As the phases of the components of the triplets are epoch de-
pendent, this phase coherency holds only if the initial epoch
corresponds to the phase of the maximum of the modula-
tion. It can be proved analytically that the phases of the
components of a modulation frequency-triplet are identical
if the modulation is pure amplitude modulation and the ini-
tial epoch corresponds to π/2 using sine terms, i.e, it is cho-
sen to be at the maximum phase of the modulation (Szeidl
2009). The phase coherency of the triplets and the symme-
try of the amplitudes of the side frequencies confirm that
the modulation of the time transformed data is basically
amplitude modulation.
Based on analytical and test results it was recently
shown by Szeidl (2009) that the phase difference between
the phases of the amplitude and phase modulations is con-
nected to the difference between the squares of the ampli-
tudes of the f0+fm and f0−fm components. If the f0+fm
component has larger amplitude than f0−fm has, then the
occurrence of the largest amplitude of the pulsation (max-
imum of the brightness maxima) precedes the occurrence

Page 7

Photometric study of the Blazhko star DM Cyg
7
5.7 5.7
5.8 5.8
Φ(f0)
Φ(f0)
2.6 2.6
2.7 2.7
Φ21
Φ21
5.4 5.4
5.5 5.5
Φ31Φ31
2 2
2.1 2.1
Φ41Φ41
4.6 4.6
4.7 4.7
-0.5 -0.5 -0.25-0.25 0 0 0.25 0.25 0.5 0.5
Φ51
Blazhko phaseBlazhko phase
Φ51
0.34 0.34
0.35 0.35
0.36 0.36
0.37 0.37
A(f0)
A(f0)
0.18 0.18
0.13 0.13
0.19 0.19
0.2 0.2
A(2f0)
A(2f0)
0.11 0.11
0.12 0.12
A(3f0)
A(3f0)
0.05 0.05
0.05 0.05
0.06 0.06
0.07 0.07
A(4f0)
A(4f0)
0.03 0.03
0.04 0.04
-0.5-0.5-0.25-0.25 0 0 0.25 0.25 0.5 0.5
A(5f0)
Blazhko phaseBlazhko phase
A(5f0)
Figure 8. Fourier amplitudes of the first 5 harmonic components
of the V light curves (left-hand panels) and the phase of the f0
pulsation frequency and the epoch independent phase differences
(right-hand panels) at 10 different phases of the Blazhko cycle.
Note, that the scales of each amplitude and phase plots are iden-
tical.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.4-0.2 0 0.2 0.4
pulsation phase
∆V [mag]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.4-0.2 0 0.20.4
pulsation phase
Figure 9. Folded light curve of the V observations (left-hand
panel) and that of the time transformed data (right-hand panel).
The times of the observations are corrected according to the phase
variation of the f0 pulsation component during the modulation
cycle in the transformed data. The inserts magnify the plots at
around the middle of the rising branch. The original data show
significant spread of the observations here due to phase modula-
tion, while in the time transformed data the phase modulation is
eliminated as the narrowness of the rising branch indicates.
of the largest delay of the maximum timings (largest pos-
itive O − C value) and the direction of the progression in
the maximum brightness–maximum light phase plot (right-
hand panel in Fig. 3) is anti-clockwise. Contrarily, if the
larger amplitude modulation components are the kf0− fm
frequencies, then the phase of the largest delay of the light
curve precedes the phase of the maximum amplitude. DM
Cyg is an example for the former case, the amplitudes of the
kf0+ fm components are larger than the amplitudes of the
kf0−fm components, the maximum of the amplitude modu-
lation precedes the phase of the largest delay of the maxima
ππππππ
3π/23π/23π/23π/23π/23π/2
2π
2π
2π
2π
2π
2π
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
phases of the triplet components [rad]
V dataV dataV dataV dataV data V data
phases of the triplet components [rad]phases of the triplet components [rad] phases of the triplet components [rad]phases of the triplet components [rad]phases of the triplet components [rad]
ππππππ
3π/23π/23π/23π/23π/23π/2
2π
2π
2π
2π
2π
2π
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
time transformed V datatime transformed V datatime transformed V data time transformed V datatime transformed V datatime transformed V data
kf0-fm
kf0+fm
kf0
0 0 0 0
0.005 0.005 0.005 0.005
0.01 0.01 0.01 0.01
1 2 3 4 5 6 7 8 9 10
order (k)order (k)order (k)order (k)
amplitudes of the triplet side components [mag]
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
amplitudes of the triplet side components [mag]amplitudes of the triplet side components [mag] amplitudes of the triplet side components [mag]
0 0 0 0
0.005 0.005 0.005 0.005
0.01 0.01 0.01 0.01
1 2 3 4 5 6 7 8 9 10
order (k)order (k) order (k)order (k)
kf0-fm
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
kf0+fm
Figure 10. Phases of the pulsation (kf0) and modulation side
lobe (kf0± fm) frequencies for the original and the time trans-
formed V data sets are shown in the top panels. The phase differ-
ences between the three components of the triplets remain close
the same in the different orders with larger deviations from these
constant values in the k = 1,2 and in the highest orders in the
original V data (left-hand panel). The phases of the components
of the triplets have nearly the same values in the different or-
ders in the time transformed data (right-hand panel). Though
the phases of the different frequency components are epoch de-
pendent, the rough constancy of the phase differences between
the triplet components at different orders holds for any epoch
both for the original and the time transformed data. According
to model calculations, the phase differences are close to zero (as
observed in the case of the time transformed data) if only ampli-
tude modulation occurs and the initial phase corresponds to the
maximum amplitude of the modulation. Bottom panels show the
amplitudes of the side lobe frequencies for the original and the
time transformed data. The kf0+fmcomponents have systemati-
cally larger amplitudes than the kf0−fmcomponents have in the
original data as shown in the bottom left panel. Contrarily, the
amplitudes of the side frequencies are very similar in each order
in the time transformed data (bottom right panel), as expected
if the modulation is pure amplitude modulation.
and the direction of the progression in the right-hand panel
in Fig. 3 is anti-clockwise.
Due to the non-sinusoidal shape of the light curve and
the residual phase modulation in the higher order pulsation
components the phases of maxima vary slightly in the time
transformed data as well. The difference between the phase
of the highest light maximum and the phase of the maximum
brightness of the largest delay during the Blazhko cycle is
around 180◦for this data set as it is shown in Fig. 11, i.e. the
largest delay of the light curve occurres when the amplitude
of the pulsation is the smallest. Fig. 11 shows the same plots
as Fig. 3 but for the time transformed data. The right-hand

Page 8

8
J. Jurcsik et al.
-0.9-0.9
-0.85-0.85
-0.5-0.5-0.25-0.25 0 0 0.25 0.25 0.5 0.5
maximum brighness
Blazhko phaseBlazhko phase
maximum brighness
-0.01-0.01
-0.005-0.005
0 0
0.005 0.005
0.01 0.01
-0.5-0.5-0.25-0.25 0 0 0.25 0.25 0.5 0.5
maximum phase
Blazhko phaseBlazhko phase
maximum phase
-0.9-0.9
-0.85-0.85
-0.01-0.01-0.005-0.005 0 0 0.005 0.005 0.01 0.01
maximum brightness
maximum phasemaximum phase
maximum brightness
Figure 11. The same figure as Fig. 3 but for the time transformed data. The phase difference between the maximum brightness and
maximum phase timings for the time transformed data is around 180◦. The maximum brightness - maximum phase plot has been
substantially compressed, the direction of the progression is denoted by an arrow. Dashed lines connect the mean values in 10 phase bins
of the modulation.
panel shows already hardly any loop structure wider than
the scatter of the data.
3.2Changes in the global physical parameters
during the Blazhko cycle
In S´ odor, Jurcsik & Szeidl (2009) we demonstrated that
from good quality multicolour photometric data the physi-
cal parameters of RRab stars can be determined with similar
accuracy as with direct Baade-Wesselink analysis, using an
inverse photometric method (IPM). We applied this method
to the BV IC light curves of MW Lyr at different phases of
its Blazhko modulation succesfully, about 1 − 2% changes
in the mean global physical parameters (L,Teff,R) were
detected during the Blazhko cycle (Jurcsik et al. 2009).
Before analysing the light curves of DM Cyg at different
phases of its modulation, first we have to determine those
parameters that definitely do not vary during the Blazhko
cycle. These are the metallicity, the mass and the distance
of the star and the dereddened standard magnitudes of the
comparison stars i.e., the zero points of the magnitude scales
used.
The [Fe/H] of DM Cyg was determined spectroscop-
ically by Suntzeff et al. (1994) and Layden (1994). The
Fourier parameters of the mean V light curve give [Fe/H]=
−0.01 using the Jurcsik & Kov´ acs (1996, Eq. 3) formula.
As the spectroscopic observations correspond to −0.16
and 0.07 [Fe/H] values on the metallicity scale of the
Jurcsik & Kov´ acs (1996) formula, atmosphere models with
[Fe/H]= 0.0 and −0.1 compositions (Castelli & Kurucz
2003) are used in the light curve modelling of DM Cyg.
The IPM finds mass values that are too large on evolu-
tionary grounds in some cases if the mass is also allowed to
vary by the fitting process. The fitting accuracy of the dif-
ferent mass solutions differs, however, only marginally. The
mean B,V,IC light curves of DM Cyg are fitted with 5.1,
4.7 and 4.0 mmag residual scatters when the mass value is
also allowed to vary. In this case the best solution is found at
M = 0.84MSun. The r.m.s. of the fixed M = 0.55MSunmass
solution B,V,IC light curve fits are 5.2, 4.6 and 4.3 mmag,
respectively. The average fitting accuracy in the three bands
decreases only 0.1 mmag, from 4.7 to 4.6 mmag, when the
mass is also allowed to be fitted. That is, the method is
rather insensitive to the value of the mass. Therefore, in or-
der to obtain reliable solutions, the IPM has been run with
0.50, 0.55 and 0.60 MSun fixed mass values for the mean
light curves of DM Cyg.
The global mean absolute physical parameters of DM
Cyg averaged over both the pulsation and the Blazhko cycles
and its distance are derived from the mean light curves using
the IP method. The results are summarized in Table 4 as-
suming different possible mass and [Fe/H] values. Note that
the zero points of the colour and magnitude scales defined
by the dereddened magnitudes of the comparison stars, are
also allowed to vary i.e, they are also fitted at this step. To
derive the distance, the apparent dereddened mean V mag-
nitude of DM Cyg has to be known as the IPM gives the
absolute mean V brightness of the star. To determine the
apparent dereddened mean magnitude of DM Cyg, the stan-

Page 9

Photometric study of the Blazhko star DM Cyg
9
Table 4. Mean physical parameters of DM Cyg derived from its mean light curves using the IP method.
[Fe/H]
fixed
M/MSun
fixed
L/LSun
Teff
R/RSun
MV
(B − V )0
(V − I)0
d [pc]
0.0 0.50
0.55
0.60
35.7 ± 1.5
38.0 ± 1.8
39.0 ± 1.4
6510 ± 55
6540 ± 40
6535 ± 40
4.66 ± 0.09
4.76 ± 0.09
4.85 ± 0.08
0.97 ± 0.05
0.90 ± 0.04
0.86 ± 0.04
0.43 ± 0.02
0.44 ± 0.01
0.42 ± 0.01
0.49 ± 0.01
0.48 ± 0.01
0.48 ± 0.01
1179 ± 23
1215 ± 21
1233 ± 21
−0.10.5538.0 ± 1.06533 ± 314.78 ± 0.090.90 ± 0.030.45 ± 0.010.48 ± 0.011210 ± 15
0.99 0.99
1.02 1.02
1.05 1.05
1.08 1.08
Amp(V)Amp(V)
-6-6
-4-4
-2-2
0 0
2 2
4 4
6 6
delta P [10-4 d]
delta P [10-4 d]
-0.3-0.3-0.3 -0.3
-0.28-0.28-0.28-0.28
-0.26-0.26-0.26-0.26
-0.24-0.24-0.24-0.24
<V>mag <V>int
(B-V) <B-V> <B>-<V>
<V>mag <V>int
(B-V) <B-V> <B>-<V>
<V>mag <V>int
(B-V) <B-V> <B>-<V>
<V>mag <V>int
(B-V) <B-V> <B>-<V>
-0.42-0.42-0.42-0.42-0.42-0.42
-0.4-0.4-0.4-0.4-0.4-0.4
-0.38-0.38-0.38-0.38-0.38-0.38
-0.36-0.36-0.36-0.36-0.36-0.36
(B-V) <B-V> <B>-<V> (B-V) <B-V> <B>-<V>
-0.36-0.36-0.36 -0.36-0.36-0.36
-0.34-0.34-0.34-0.34-0.34-0.34
-0.32-0.32 -0.32-0.32-0.32-0.32
-0.3-0.3-0.3-0.3-0.3-0.3
0 0 0 0 0 0 0.5
Blazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phase
1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2
(V-I) <V-I> <V>-<I>
0.5 0.5 0.5 0.5 0.5
(V-I) <V-I> <V>-<I> (V-I) <V-I> <V>-<I> (V-I) <V-I> <V>-<I> (V-I) <V-I> <V>-<I> (V-I) <V-I> <V>-<I>
4.66 4.66 4.66 4.66 4.66 4.66
4.67 4.67 4.67 4.67 4.67 4.67
R/RSun
M=0.50 MSun
M=0.50 MSun
M=0.50 MSun
M=0.50 MSun
M=0.50 MSun
M=0.50 MSun
R/RSun
R/RSun
R/RSun
R/RSun
R/RSun
2.796 2.796 2.796 2.796 2.796 2.796
0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92
2.798 2.798 2.798 2.798 2.798 2.798
2.8 2.8 2.8 2.8 2.8 2.8
log g log glog glog glog glog g
0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94
0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98
35.8 35.8 35.8 35.8 35.8 35.8
MV
MV
MV
MV
MV
MV
MV
MV
MV
MV
MV
MV
35.6 35.6 35.6 35.6 35.6 35.6
35.7 35.7 35.7 35.7 35.7 35.7
L/LSun
L/LSun
L/LSun
L/LSun
L/LSun
L/LSun
6500 6500 6500 6500 6500 6500
6505 6505 6505 6505 6505 6505
6510 6510 6510 6510 6510 6510
6515 6515 6515 6515 6515 6515
0 0 0 0 0 0 0.5
Blazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phase
1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2
Teff
0.5 0.5 0.5 0.5 0.5
Teff
Teff
Teff
Teff
Teff
4.76 4.76 4.76 4.76 4.76 4.76
4.77 4.77 4.77 4.77 4.77 4.77
M=0.55 MSun
M=0.55 MSun
M=0.55 MSun
M=0.55 MSun
M=0.55 MSun
M=0.55 MSun
2.82 2.82 2.82 2.82 2.82 2.82
2.822 2.822 2.822 2.822 2.822 2.822
2.824 2.824 2.824 2.824 2.824 2.824
0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84
0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86
0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88
0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
37.9 37.9 37.9 37.9 37.9 37.9
6550 6550 6550 6550 6550 6550
38 38 38 38 38 38
38.1 38.1 38.1 38.1 38.1 38.1
6535 6535 6535 6535 6535 6535
6540 6540 6540 6540 6540 6540
6545 6545 6545 6545 6545 6545
0 0 0 0 0 0 0.5
Blazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phase
1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5
4.85 4.85 4.85 4.85 4.85 4.85
4.86 4.86 4.86 4.86 4.86 4.86
4.87 4.87 4.87 4.87 4.87 4.87
M=0.60 MSun
M=0.60 MSun
M=0.60 MSun
M=0.60 MSun
M=0.60 MSun
M=0.60 MSun
2.842 2.842 2.842 2.842 2.842 2.842
2.844 2.844 2.844 2.844 2.844 2.844
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8
0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82
0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84
0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86
39.2 39.2 39.2 39.2 39.2 39.2
39.3 39.3 39.3 39.3 39.3 39.3
39.4 39.4 39.4 39.4 39.4 39.4
6525 6525 6525 6525 6525 6525
6530 6530 6530 6530 6530 6530
6535 6535 6535 6535 6535 6535
6540 6540 6540 6540 6540 6540
0 0 0 0 0 0 0.5
Blazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phaseBlazhko phase
1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5
Figure 12. Variations of the observed mean (left-hand panels) and derived parameters of DM Cyg during the Blazhko cycle. Circles and
crosses denote the results when the IP method use Liu’s template radial velocity curve, and when the Vradcurve is calculated from the
IClight curve, respectively. No significant, systematic difference between the results can be seen depending on the choice of the initial
radial velocity templates. The results of the IP method for fixed 0.50, 0.55 and 0.60 M/MSunmass values show the same variations in
each parameters. Only the absolute mean values of the parameters are different, according to the data given in Table 4. These results
prove that the detected changes of the physical parameters during the Blazhko cycle of DM Cyg are not sensitive to the settings of the
IP method, and are independent of the uncertainties of the values of the mean global parameters. See further details in the text.
dard V magnitude of the comparison stars and the interstel-
lar absorption AV should have to be known. Unfortunately,
no standard magnitudes of the comparison stars have been
published. We estimate the average of the dereddened V
brightnesses of the two comparison stars to be 11.333 mag
transformed from their Tycho BT, and VT magnitudes (ESA
1997) and using AV = 3.14E(B−V ) = 0.558 interstellar ab-
sorption value according to the Schlegel et al. (1998) maps.
It has to be emphasized, however, that the uncertainties of
the standard V magnitudes of the comparison stars and the
interstellar absorption value affect only the distance esti-
mate. Neither any other parameter nor their detected varia-
tions depend on the choice of the dereddened V magnitudes
of the comparison stars.
The uncertainties of the estimates of the absolute phys-
ical parameters listed in Table 4 correspond to the standard
deviations of the results of running the IP code with 16
different settings. These settings are the combinations of:
i) B, V and IC, or V , B − V and V − IC data are used;
ii) the Vrad curve is defined by Liu’s template or it is cal-

Page 10

10
J. Jurcsik et al.
culated from the IC light curve; iii) 2 different weights of
the initial Vrad curve are used; iv) 2 different values of the
∆A(Vrad)/∆VAmp ratio valid for Blazhko variables are ap-
plied (see further details in S´ odor, Jurcsik & Szeidl 2009;
Jurcsik et al. 2009). However, as it can be seen from the
data listed in Table 4, the true uncertainties of the mean
global parameters are larger than the estimated errors of
the solutions obtained for fixed mass and metallicity values.
The true possible parameter space comprises the full range
of the solutions for the possible mass and metallicity ranges.
Nevertheless, we are focusing on the variation in the mean
physical parameters and not on their absolute values, and it
is shown in the rest of this section that the uncertainties of
the mean global physical parameters have no effect on their
variation during the Blazhko cycle.
Though the amplitude of the modulation of DM Cyg is
only about one tenth of that of MW Lyr, we have applied
the IPM to the light curves in 10 different phases of the mod-
ulation in order to decide whether there are any detectable
changes in the mean global parameters of DM Cyg during
its Blazhko period. The distance and the mass are fixed to
their possible values given in Table 4 assuming [Fe/H]= 0.0
metallicity. The zero points of the magnitude scales are also
fixed correspondingly to the solution obtained for the mean
light curves.
Fig. 12 shows the results for the solutions using M =
0.50MSun, d = 1179 pc; M = 0.55MSun, d = 1215 pc and
M = 0.60MSun, d = 1233 pc mass and distance values. In
the left-hand panels, the variations in the observed mean
quantities are plotted: total pulsation amplitude in V band,
the variation of the pulsation period derived from the phase
variation of the f0 pulsation frequency and different pulsa-
tion averages of the magnitudes and colours. The right-hand
panels show the quantities derived from the IP method for
three possible mass/distance combinations: the mean val-
ues of the radius, the surface gravity, the absolute visual
brightness averaged by magnitude and intensity units, the
luminosity and the effective temperature. The different com-
binations of the mass and distance give very similar results,
the amplitudes and phases of the detected changes in the
mean physical parameters during the Blazhko cycle hardly
change, only their averages over the Blazhko cycle are dif-
ferent, corresponding to their respective values given in Ta-
ble 4. If atmosphere models with [Fe/H]=−0.1 are used, the
results on the variations of the physical parameters during
the Blazhko cycle are not changed.
In order to get a real estimate of the uncertainties of
the derived quantities in different phases of the modulation,
the IP code has been run with 16 different settings for each
data set again. Note that the zero points of the colour and
magnitude scales defined by the dereddened magnitudes of
the comparison stars, are also fixed now. These magnitudes
were, however, allowed to vary i.e., they were also fitted
when the mean global parameters were determined. This
explains why the scatter of the results for the 16 different
setting of the IP method is much smaller in the different
phases of the modulation than the scatter of the results for
the mean light curves.
The different intensity and magnitude averages of the
V light curve and the colour curves show hardly any vari-
ations, the amplitudes of their changes is only 0.002-0.004
mag. The V , and B − V averages show some small system-
atic variations of similar character like these averages in MW
Lyr, but the V − I averages show only scatter. In spite of
that the observed mean magnitudes and colours vary only
slightly during the Blazhko cycle in DM Cyg, by the aid
of the IP method 0.3% and 7 K systematic changes in the
mean luminosity and temperature of the star could be de-
tected during its Blazhko cycle. The phase relation of these
variations are the opposite, similarly as it was detected in
MW Lyr. Both stars are the most luminous and the coolest
at around the largest amplitude phase of the modulation.
The IP method calculates the mean physical parame-
ters as their mathematical averages over the pulsation cycle.
F(Φ) =< F(ϕ) >, where Φ and ϕ denote modulation and
pulsation phases, respectively, and <> denotes averaging by
ϕ for the whole pulsation cycle. For the luminosity, it means
that the values plotted in Fig. 12 correspond to:
L(Φ) =< L(ϕ) >=< 4πσR(ϕ)2Teff(ϕ)4>,
as the IP method satisfies the Stephan-Boltzmann law in
each phase of the pulsation (ϕ).
Carney et al.(1992)showed
Wesselink analysis results that, even when the equilibrium
luminosity and radius of pulsating variables equal with the
arithmetic means of their variations over the pulsation cy-
cle, the equilibrium temperature Teq is not the same as the
mean temperature < Teff > of the star. The equilibrium
temperature is defined then as
Teq = (Leq/4πσR2
= (< L(ϕ) > /4πσ < R(ϕ) >2)1/4.
Consequently, the relation between the arithmetic means of
the physical parameters shown in Fig. 12 deviates to some
extent from the theoretical Stefan-Boltzmann law.
The 0.13% variation in the mean radius is in good agree-
mant with the 0.17% changes of the pulsation period (the
pulsation equation requires ∆P/P ≈ 3/2∆R/R). The pul-
sation period changes are determined from the phase differ-
ences of the f0 pulsation frequency in different phases of the
modulation, but when deriving the radius variation the IP
method does not utilize this information in any way. Con-
sequently, the determined period and radius variations are
completely independent quantities. The fact, that they show
such a good agreement is a great support both to the IP
method, and to our interpretation of the phase modulation
as variations in the pulsation period.
Comparing the results obtained for DM Cyg with that
of MW Lyr (Fig. 14 in Jurcsik et al. 2009) the similar be-
haviour of the two stars are conspicuous. The only difference
is in the detected amplitudes of the variations of the differ-
ent parameters, each being about one tenth of their detected
amplitudes in MW Lyr, in accordance with the different am-
plitudes of the modulation of the two stars.
fromdirectBaade-
eq)1/4= (L(Φ)/4πσR(Φ)2)1/4
4THE PHOTOGRAPHIC AND
PHOTOELECTRIC DATA
The photographic light curve of DM Cyg was shown in Hurta
(2009). Although the Konkoly photographic data are very
sparse, the observations covered 24 years with a 13-year gap,
we tried to analyse this data set also searching for any sign
of light curve modulation. No indication of light curve mod-
ulation has been found in any of the two parts of the pho-
tographic data, but there is some hint that modulation fre-

Page 11

Photometric study of the Blazhko star DM Cyg
11
0.01
0.02
0.03
0.04
k = 1
0.01
0.02
0.03
0.04
k = 2
0.01
0.02
0.03
0.04
kf0-fm
kf0
kf0+fm
k = 3
0
0.2
0.4
0.6
0.8
1
-fm
0
+fm
spectral window
frequency
Figure 13.
light curve (after the removal of the pulsation signal) in the vicin-
ity of the f0,2f0 and 3f0 pulsation frequencies. The residuals
show signals at around f0+fm, 2f0+fm and 3f0+fm frequen-
cies with 0.02 − 0.03 mag amplitude. The f0 and fm frequencies
detected in the photographic data are 2.3817564 and 0.0940, re-
spectively. Bottom panel shows the spectral window of the pho-
tographic observations.
Amplitude spectrum of the residual photographic
11.25
11.3
11.35
-0.4-0.2
Blazhko phase
0 0.2 0.4
Bpg [mag]
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
-0.4-0.2
Blazhko phase
0 0.2 0.4
max phase
Figure 14. Maximum brightness and phase of the Bpg
11.5 mag brightness values on the rising branch (0.2 mag below
maximum brightness) for the photographic observations phased
with the supposed modulation period (10.636 days) are shown in
the left-hand and right-hand panels, respectively. Data from the
two parts of the photographic observations are shown with dif-
ferent symbols. According to these plots, there is some indication
that the maximum brightness and the phase of the rising branch
of DM Cyg varied with small amplitude during the time of the
photographic observations as well.
=
quency components appear at kf0 + fm,(k = 1,2,3) posi-
tions in the residual spectrum of the complete photographic
data set (see Fig. 13). The highest modulation peaks are not
exactly at the same separations for f0+ fm, 2f0+ fm and
3f0+ fm. The best solution which gives the smallest resid-
ual for the photographic data can be gained with 2.3817564
c/d (0.41985822 d) and 0.0940 c/d (10.636 d) pulsation and
modulation frequencies, respectively.
Fig. 14 plots the maximum brightness and the phase
-0.05
0
0.05
0.1
0.15
0.2
15000 20000 25000 30000 35000 40000 45000 50000 55000
JD - 2400000
O - C [day]
pgpeCCD
Figure 15. The O −C diagram of DM Cyg. Literature data are
from the GEOS database (dots), the most deviant data are omit-
ted. The Konkoly photographic, photoelectric and CCD normal
maximum timings listed in Table 2 are plotted by squares, the
time intervals of the Konkoly observations are indicated by hori-
zontal lines in the top of the figure. The O − C indicates steady
period increase during the ∼hundred years of the observations.
of the rising branch at Bpg = 11.5 mag brightness values
phased with the supposed 10.636 d modulation period. The
phase of a given magnitude on the rising branch can be de-
termined with higher accuracy than the phase of the max-
imum, therefore we use this quantity to measure the am-
plitude of the phase modulation component in the photo-
graphic data. Data belonging to the two parts of the obser-
vations are denoted by different symbols in Fig. 14. Accord-
ing to these plots the amplitudes of the amplitude and phase
modulations were not larger at the time of the photographic
observations than today.
The photographic data were also analysed taking into
account the period decrease of the pulsation that took place
during the time interval of the observations, but the results
were only marginally different from that obtained from the
original photographic data.
The photoelectric observations comprise data only from
four nights which does not allow to check the light curve
modulation. The photoelectric data were therefore used only
to determine one normal maximum timing listed in Table 2.
The normal maximum timings of the Konkoly obser-
vations given in Table 2 together with literature data col-
lected in the GEOS database are plotted in Fig. 15. Based
on the GEOS data Le Borgne et al. (2007) determined β =
0.091dMyr−1period increase rate for DM Cyg. The addi-
tion of the Konkoly data does not modify this result.
From the analyses of the CCD and photographic data of
DM Cyg we conclude that the period changes of the pulsa-
tion and modulation have the opposite direction, when the
pulsation period of DM Cyg was shorter by 0.000005 d then
the modulation period was 0.066 d longer than today. These
values correspond todPB
dP0= −13200 or
riod change rates. However, these data have to be taken with
caution partially because the uncertainties of the results de-
rived from the photographic data and also because pulsation
and modulation period changes of those Blazhko variables
where the periods could be determined for several epochs
show that there is no strict relation between the two periods,
for some time intervals even the sign of the period change
dPB
PB/dP0
P0= −523 pe-

Page 12

12
J. Jurcsik et al.
ratio can change (e.g., in RV UMa Hurta et al. 2008). There-
fore, the period change ratios defined by only two-epoch data
should be missleading in some cases. We also have to note,
however, that the complex period change behaviour of the
modulation is usually connected to complex changes of the
pulsation period and of the properties of the modulation, too
(e.g., in RR Gem and XZ Cyg S´ odor, Szeidl & Jurcsik 2007;
LaCluyz´ e et al. 2004). Probably, the steady period change
of the pulsation of DM Cyg is accompanied with steady pe-
riod change of its modulation and with the stability of its
modulation properties.
5 SUMMARY
In the present paper the light curve modulation of DM Cyg,
a fundamental mode RR Lyrae variable has been investi-
gated. Most of the results are very similar to those found
in the analysis of MW Lyrae, a large modulation amplitude
Blazhko variable, but on a significantly smaller scale. The
0.07 mag amplitude of maximum brightness variation of DM
Cyg is only 10% of the modulation amplitude of MW Lyr,
accordingly, all the changes detected in DM Cyg during its
Blazhko period are about 10% of the detected changes in
MW Lyr.
The phase and amplitude relations of the amplitude
and phase modulation components are similar in DM Cyg
and MW Lyr. The variations of the mean magnitudes and
colours in DM Cyg where they can be detected are also sim-
ilar to their counterparts in the MW Lyr data. Therefore, it
is not surprising that the derived changes in the mean global
physical parameters of DM Cyg resemble the changes found
in the mean parameters of MW Lyr only on a much reduced
scale.
The only notable difference between the character of
the modulations of the two stars is found when the changes
of the Fourier parameters of the light curves during the
Blazhko cycle is investigated. While in the case of MW Lyr
the phases of 2f0, and 3f0 change in line with the phase vari-
ation of f0(i.e, there is no significant changes detected in the
ϕ(fk1) phase differences), when analysing the light curves of
DM Cyg 90◦phase differences between the phase variation
of the f0 pulsation frequency and the phase variations of
the higher order pulsation components are found. In spite of
this, the data transformation that corrects the times of the
observations taking into account the phase variation of the
f0 pulsation frequency which completely separated the am-
plitude and phase modulation components of the light curve
modulation of MW Lyr seems to work also on the light curve
of DM Cyg.
The modulation behaviour of other Blazhko variables
can differ, however, more significantly from the modulations
of DM Cyg and MW Lyr. There are Blazhko stars, where the
phase relation of the amplitude and phase modulation com-
ponents contrast with the phase relations of DM Cyg and
MW Lyr. Also, the detected changes in the mean colours
may vary differently e.g, in SS Cnc (Jurcsik et al. 2006).
Therefore, we cannot draw major conclusions from the sim-
ilarity of the results obtained for DM Cyg and MW Lyr be-
fore analysing other Blazhko variables of different character
on a similar way.
6ACKNOWLEDGMENTS
The financial support of OTKA grants T-068626 and T-
048961 is acknowledged. We wish to thank K. Ol´ ah for ob-
taining the photoelectric observations of DM Cyg. Zs.K. is
a grantee of the Bolyai J´ anos Scholarship of the Hungarian
Academy of Sciences.
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arXiv:0904.4129v1 [astro-ph.SR] 27 Apr 2009
Mon. Not. R. Astron. Soc. 000, 1–12 (2009)Printed 27 April 2009(MN LATEX style file v2.2)
An extensive photometric study of the Blazhko RR Lyrae
star DM Cyg⋆
J. Jurcsik1Zs. Hurta1,2,´A. S´ odor1, B. Szeidl1, I. Nagy2, K. Posztob´ anyi4,
M. V´ aradi1,3, K. Vida1,2, B. Belucz2, I. D´ ek´ any1, G. Hajdu2, Zs. K˝ ov´ ari1, E. Kun5
1Konkoly Observatory of the Hungarian Academy of Sciences, H-1525 Budapest PO Box 67, Hungary
2E¨ otv¨ os University, Dept. of Astronomy, H-1518 Budapest PO Box 49, Hungary
3Observatoire de Geneve, Universite de Gen` eve, CH-1290, Sauverny, Switzerland
4AEKI, KFKI Atomic Energy Research Institute, Thermohydraulic Department, H-1525 Budapest 114, PO Box 49, Hungary
5Department of Experimental Physics and Astronomical Observatory, University of Szeged, 6720 Szeged, D´ om t´ er 9, Hungary
Accepted 2009 December 15. Received 2009 December 14; in original form 2009 January 22
ABSTRACT
DM Cyg, a fundamental mode RRab star was observed in the 2007 and 2008
seasons in the frame of the Konkoly Blazhko Survey. Very small amplitude light curve
modulation was detected with 10.57 d modulation period. The maximum brightness
and phase variations do not exceed 0.07 mag and 7 min, respectively. In spite of the
very small amplitude of the modulation, beside the frequency triplets characterizing
the Fourier spectrum of the light curve two quintuplet components were also identified.
The accuracy and the good phase coverage of our observations made it possible to
analyse the light curves at different phases of the modulation separately. Utilizing the
IP method (S´ odor, Jurcsik and Szeidl, 2009) we could detect very small systematic
changes in the global mean physical parameters of DM Cyg during its Blazhko cycle.
The detected changes are similar to what we have already found for a large modulation
amplitude Blazhko variable MW Lyrae. The amplitudes of the detected changes in the
physical parameters of DM Cyg are only about 10% of that what have been found in
MW Lyr. This is in accordance with its small modulation amplitude being about one
tenth of the modulation amplitude of MW Lyr.
The pulsation period of DM Cyg has been increasing by a rate of β = 0.091dMyr−1
during the hundred-year time base of the observations. Konkoly archive photographic
observations indicate that when the pulsation period of the variable was shorter by
∆ppuls= 5·10−6d the modulation period was longer by ∆pmod= 0.066 d than today.
Key words:
DM Cyg – techniques: photometric – methods: data analysis –
stars: variables: other – stars: horizontal branch – stars: individual:
1INTRODUCTION
Utilizing our full access to an automatic 60 cm telescope we
have obtained extended multicolour observations of many
fundamental mode RR Lyrae variables showing light curve
modulation (the Blazhko effect) during the past five years.
Detailed analyses of some of our targets were already pub-
lished in Jurcsik et al. (2005, 2006, 2008, 2009). Our obser-
vations are the first multicolour photometric data which are
accurate and dense enough to allow not only to determine
⋆Based on observations collected with the automatic 60 cm tele-
scope of Konkoly Observatory, Budapest, Sv´ abhegy
the frequencies appearing in the Fourier spectra of the light
curves, but also to find changes in the global mean physical
parameters of the stars (L,Teff,R) in different phases of
the modulation. In order to extract changes in the phys-
ical parameters exclusively from multicolour photometric
data we developed an inverse photometric Baade-Wesselink
method (IPM; S´ odor, Jurcsik & Szeidl 2009). The IPM was
successfully applied to determine changes in the mean global
physical parameters of MW Lyr during its Blazhko cycle in
Jurcsik et al. (2009).
DM Cyg, a short period (P = 0.42 d) RRab star was
selected to be observed in the frame of the Konkoly Blazhko
Survey because it had been announced to show phase modu-

Page 14

2
J. Jurcsik et al.
Table 1. Konkoly photometric observations of DM Cyg.
HJD
2400000+
mag∗
detectorfilter
54265.45479
54265.46045
54265.46611
...
−0.249
−0.404
−0.501
...
CCD
CCD
CCD
...
V
V
V
...
∗Relative magnitudes for the CCD and photoelectric data.
Table 2. Normal maximum timings derived from the Konkoly
photographic, photoelectric and CCD observations.
JD Normal maximum timingsobs.
2427667 − 2429226
2434186 − 2436373
2443747 − 2444133
2454265 − 2454407
2454597 − 2454711
2429078.482
2435342.347
2443777.333
2454334.3467
2454661.4210
pg
pg
pe
CCD
CCD
lation with a period of 26 d by Lysova & Firmanyuk (1980),
but neither the NSVS data (Wo´ zniak et al. 2004) nor the
complete data set of its maximum timings show phase mod-
ulation with this period (S´ odor & Jurcsik 2005).
Though DM Cyg is a relatively bright (V = 10−11 mag)
RR Lyrae star, complete, accurate, multicolour light curve
of its pulsation has never been published. Its period change
was, however, regularly monitored by different groups of ob-
servers. The GEOS database1lists 260 maximum times of
DM Cyg between 1900 and 2008.
In the present paper we publish our extended CCD ob-
servations of DM Cyg and the results of the analysis of its
light curve modulation. Archive photographic and photo-
electric Konkoly data are also processed.
2 DATA
CCD observations were obtained with the automated 60
cm telescope of the Konkoly Observatory, Sv´ abhegy, Bu-
dapest equipped with a Wright Instruments 750 × 1100
CCD camera and BV IC filters. Measurements were taken
on 81 nights between July 2007 and Sept 2008. About
3100 data points in each band were gathered. Exposition
times were 200, 60 and 40 sec or a bit longer depend-
ing on the sky transparency in the BV IC bands, respec-
tively. Data reduction was performed using standard IRAF2
packages. Aperture photometry of DM Cyg (21:21:11.548
+32:11:28.71) and several neighbouring stars were carried
out in order to check the stability of the photometry and
the constancy of the comparison stars. The relative mag-
nitudes of DM Cyg measured to the mean magnitudes of
C1=GSC2.2 N0330220980 (21:21:30.731 +32:13.05.78), and
1http://dbrr.ast.obs-mip.fr/maxRR.html
2IRAF is distributed by the National Optical Astronomy Obser-
vatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation.
-1
-0.75
-0.5
-0.25
0
0.25
54300 54400 54500 54600 54700
delta V
JD-2400000
A
-1
-0.75
-0.5
-0.25
0
0.25
-0.5-0.25 0 0.25 0.5
deltaV
pulsation phase
B
-1
-0.75
-0.5
-0.25
0
0.25
-0.5-0.25 0 0.25 0.5
deltaV
modulation phase
C
Figure 1. Delta V magnitudes versus Julian Date, data phased
with the 0.419863d pulsation and the 10.57d modulation periods
are shown in panels A, B and C, respectively.
C2=GSC2.2 N03302207371 (21:20:59.483 +32:13.01.63) are
used in the analysis. The Tycho B and V magnitudes of the
comparison stars C1 and C2 are BT = 12.705, VT = 11.799,
and BT = 13.081,VT = 12.150, respectively (Hog et al.
2000). Second order extinction correction of the data were
applied in the B band. Relative magnitudes are transformed
to standard BV IC magnitudes. More details on the reduc-
tion procedure are given in Jurcsik et al. (2008).
Archive photoelectric and photographic data obtained
with the 60 cm telescope in 1978 and with a 16 inch astro-
graph between 1934 and 1958 are also utilized. Photoelectric
observations were obtained on 4 nights, the BV magnitudes
were measured relative to GSC2.2 N03302207371. The pho-
tographic measurements comprise data from 40 nights, the
plates were evaluated using GSC2.2 B magnitudes of the
surrounding stars.
Table 1 shows a sample of the Konkoly photometric
data of DM Cyg. All the photometric observations are avail-
able in the online version of the journal as Supplementary
Material.
Seasonal normal maximum timings from each data set
have been determined for five epochs, these data are listed
in Table 2.
3RESULTS
The accurate CCD observations reveal that the light curve of
DM Cyg is not stable. It shows small amplitude modulation,
the variations in maximum brightness and phase are only
about 0.07 mag and 0.005 d (7 min, 0.012 pulsation phase),
respectively.
The V light curve of DM Cyg and the results obtained
folding the data with the pulsation and modulation periods
are shown in the three panels of Fig. 1. The phased resid-
ual light curve after the removal of the mean pulsation light
variation is given in Fig. 2. This figure shows the differences

Page 15

Photometric study of the Blazhko star DM Cyg
3
-0.9-0.9
-0.85-0.85
-0.5-0.5-0.25-0.25 0 0 0.25 0.25 0.5 0.5
maximum brightness
Blazhko phaseBlazhko phase
maximum brightness
-0.01-0.01
-0.005 -0.005
0 0
0.005 0.005
0.01 0.01
-0.5 -0.5-0.25 -0.25 0 0 0.25 0.25 0.5 0.5
maximum phase
Blazhko phase Blazhko phase
maximum phase
-0.9 -0.9
-0.85 -0.85
-0.01-0.01-0.005 -0.005 0 0 0.005 0.005 0.01 0.01
maximum brightness
maximum phasemaximum phase
maximum brightness
Figure 3. Maximum V brightness and maximum phase values phased according to the Blazhko period are plotted in the top-left and
bottom-left panels, respectively. Their typical errors are 0.005 mag and 0.002-0.004 phase. The right-hand panel shows the maximum
brightness vs maximum phase data. The progression of data during the Blazhko cycle in this diagram is anti-clockwise. Dashed lines
connect the mean values in 10 phase bins of the modulation.
-0.08-0.08 -0.08
-0.06-0.06-0.06
-0.04-0.04-0.04
-0.02-0.02-0.02
0 0 0
0.02 0.02 0.02
0.04 0.04 0.04
0.06 0.06 0.06
0.08 0.08 0.08
-0.5-0.5 -0.5-0.25-0.25-0.25 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5
Vresidual
pulsation phasepulsation phasepulsation phase
Vresidual
Vresidual
-0.5-0.5-0.5-0.25-0.25-0.25 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5
Vresidual
pulsation phasepulsation phasepulsation phase
Vresidual
Vresidual
Figure 2. Residual V light curve of DM Cyg after removing the
pulsation components from the data according to the light curve
solution given in Table 3. Residuals of the smallest (stars) and
the largest (diamonds) amplitude phases are shown with large
symbols in the left-hand and right-hand panels, respectively. Note
that the residuals have larger amplitude at around the middle of
the rising branch of the pulsation (phase = −0.1) than around
pulsation maxima (phase = 0) due to phase modulation.
of the amplitude of the modulation at different phases of the
pulsation. The well observable amplitude of the modulation
is concentrated to a narrow phase range of the pulsation
around minimum, rising branch and maximum. A similar
behaviour has been already found in RR Gem and SS Cnc,
in two other small modulation amplitude Blazhko variables
(Jurcsik et al. 2005, 2006). The amplitude of the the resid-
ual variation is the largest at around the middle of the rising
branch of the pulsation light curve (phase = −0.1) indicat-
ing phase modulation of the rising branch as well. There is a
difference between the phase of the amplitude modulation of
the maximum of the light curve and the phase of the phase
modulation of the same feature. In particular the maximum
positive displacement of the timings of the maximum bright-
ness precedes the occurrence of the brightest maximum by
about 0.25 Blazhko phase as it is shown Fig. 3.
The elements of the pulsation and the modulation are:
Tmax puls= 2454312.514 [HJD] + 0.419863 · Epuls, (1)
and
Tmax Bl= 2454312.514 [HJD] + 10.57 · EBl.(2)
Epulsand EBldenote the epoch number of the pulsation and
modulation cycles, respectively.
The pulsation and modulation periods correspond to
the mean values of the frequencies of the light curve so-
lutions of the B,V and IC data using ‘locked’ frequency
solutions of the kf0 pulsation and kf0± fm, and fm modu-
lation frequency components (i.e., the modulation side lobe
components are at the positions of the linear combination
frequencies).
Data analysis was performed using the different appli-
cations of the MUFRAN package (Koll´ ath 1990), a linear
combination fitting program developed by´A. S´ odor, and
the linear and nonlinear curve fitting abilities of gnuplot3.
3http://www.gnuplot.info/