Does Kepler unveil the mystery of the Blazhko effect? First detection of period doubling in Kepler Blazhko RR Lyrae stars

Abstract

The first detection of the period doubling phenomenon is reported in the Kepler RR Lyrae stars RR Lyr, V808 Cyg and V355 Lyr. Interestingly, all these pulsating stars show Blazhko modulation. The period doubling manifests itself as alternating maxima and minima of the pulsational cycles in the light curve, as well as through the appearance of half-integer frequencies located halfway between the main pulsation period and its harmonics in the frequency spectrum. The effect was found to be stronger during certain phases of the modulation cycle. We were able to reproduce the period doubling bifurcation in our nonlinear RR Lyrae models computed by the Florida-Budapest hydrocode. This enabled us to trace the origin of this instability in RR Lyrae stars to a resonance, namely a 9:2 resonance between the fundamental mode and a high-order (9th) radial overtone showing strange-mode characteristics. We discuss the connection of this new type of variation to the mysterious Blazhko effect and argue that it may give us fresh insights to solve this century-old enigma. Comment: 10 pages, 12 figures and 1 table, accepted for publication in MNRAS

Full text

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arXiv:1007.3404v1 [astro-ph.SR] 20 Jul 2010
Mon. Not. R. Astron. Soc. 000, 1–10 (2010)Printed 21 July 2010(MN LATEX style file v2.2)
Does Kepler unveil the mystery of the Blazhko effect? First
detection of period doubling in Kepler Blazhko RR Lyrae
stars
R. Szab´ o1⋆, Z. Koll´ ath1, L. Moln´ ar1, K. Kolenberg2, D. W. Kurtz3,
S. T. Bryson4, J. M. Benk˝ o1, J. Christensen-Dalsgaard5, H. Kjeldsen5,
W. J. Borucki4, D. Koch4, J. D. Twicken6, M. Chadid7, M. Di Criscienzo8,
Y-B. Jeon9P. Moskalik10, J. M. Nemec11, J. Nuspl1
1Konkoly Observatory of the Hungarian Academy of Sciences, Konkoly Thege Mikl´ os ´ ut 15-17, H-1121 Budapest, Hungary
2Institut f¨ ur Astronomie, University of Vienna, T¨ urkenschanzstrasse 17, A-1180 Vienna, Austria
3Jeremiah Horrocks Institute of Astrophysics, University of Central Lancashire, Preston PR1 2HE, UK
4NASA Ames Research Center, MS 244-30, Moffet Field, CA 94035, USA
5Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
6SETI Institute/NASA Ames Research Center, Moffett Field, CA 94035
7Observatoire de la Cˆ ote d’Azur, Universit´ e Nice, Sophia-Antipolis, UMR 6525, parc Valrose, 06108, Nice Cedex 02, France
8INAF-Osservatorio Astronomico di Roma, via Frascati 33, Monte Porzio Catone, Roma, Italy
9Korea Astronomy and Space Institute, Daejeon, 305-346, Korea
10Copernicus Astronomical Center, ul. Bartycka 18, 00-716, Warsaw, Poland
11Department of Physics & Astronomy, Camosun College, Victoria, British Columbia, Canada
Accepted; Received; in original form 2010 April
ABSTRACT
The first detection of the period doubling phenomenon is reported in the Kepler
RR Lyrae stars RR Lyr, V808 Cyg and V355 Lyr. Interestingly, all these pulsating
stars show Blazhko modulation. The period doubling manifests itself as alternating
maxima and minima of the pulsational cycles in the light curve, as well as through
the appearance of half-integer frequencies located halfway between the main pulsa-
tion period and its harmonics in the frequency spectrum. The effect was found to be
stronger during certain phases of the modulation cycle. We were able to reproduce
the period doubling bifurcation in our nonlinear RR Lyrae models computed by the
Florida-Budapest hydrocode. This enabled us to trace the origin of this instability
in RR Lyrae stars to a resonance, namely a 9:2 resonance between the fundamental
mode and a high-order (9th) radial overtone showing strange-mode characteristics. We
discuss the connection of this new type of variation to the mysterious Blazhko effect
and argue that it may give us fresh insights to solve this century-old enigma.
Key words: Kepler – instabilities – stars: oscillations – stars: variables: RR Lyr –
stars individual: RR Lyrae – stars individual: V808 Cyg – stars individual: V355 Lyr.
1INTRODUCTION
The unprecedented power of the Kepler1space telescope in
terms of precision and continuity is poised to deliver ma-
jor breakthroughs in exoplanet science (Borucki et al. 2010)
and stellar photometry (Gilliland et al. 2010a) allowing ex-
ploration of territories never tested before. New instruments
often reveal surprising new phenomena and lead to new in-
⋆E-mail: rszabo@konkoly.hu
1http://kepler.nasa.gov
sights in (astro)physical problems. Such an unforeseen fea-
ture, period doubling2(hereafter PD) and the corresponding
half-integer frequencies (hereafter HIFs) were reported by
Kolenberg et al. (2010a) in the Kepler Q1 data (34 days) of
RR Lyr, the prototype, brightest Blazhko-type RR Lyrae
star in the sky.
It has been well known for decades that high-
2The period doubling phenomenon is not to be confused with
double-mode pulsation, where two radial modes (of low order)
are excited simultaneously.

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2R. Szab´ o, Z. Koll´ ath, L. Moln´ ar, et al.
luminosity RV Tauri variables show alternating deep and
shallow minima in their light and radial velocity curves.
Buchler & Kov´ acs (1987) and Kov´ acs & Buchler (1988) car-
ried out the first systematic search of irregular oscillations
in radiative and strongly dissipative Pop. II (W Vir) models.
They demonstrated that the pulsations in those models un-
dergo a Feigenbaum cascade of period doubling bifurcations
by changing the control parameter (effective temperature).
That is to say the instability develops from strict periodic
pulsation to period-two, period-four etc., oscillations result-
ing in low dimensional chaos.
Moskalik & Buchler
Buchler & Moskalik (1992)
bifurcation, as well, in purely radiative Cepheid and BLHer
model sequences. By changing the control parameter (Teff),
their weakly dissipative Pop. I. Cepheid models showed
the onset of period doubling and a subsequent reversion to
period-one oscillations instead of further period-doubling
episodes, in contrast with the more dissipative models.
Nonlinear stable periodic pulsations (limit cycles) can
be made to ’period double’ through the destabilization of
either a thermal (real) mode or an additional (complex) vi-
brational mode. Moskalik & Buchler (1990) could trace the
origin of the PD to be a destabilized low-lying vibrational
overtone and that the coupling occurs through an internal
resonance of the type (2n+1)ω0 ≈ 2ωkwhere n in an integer
(1 or 2) and the subscripts 0 and k refer to the fundamen-
tal and the kth overtone modes, respectively. In this case the
parametric instability of an overtone pulsation mode in half-
integer resonance opens up an additional dimension which
allows the limit cycle to period double.
In this paper we describe our discovery of the period
doubling phenomenon in three RR Lyrae variables observed
by the Kepler space telescope. One of them is RR Lyr
(KIC7198959, Kepler mag: 7.9) the prototype of its class.
V808 Cyg (KIC4484128) and V355 Lyr (KIC7505345), the
two other RRab stars are of much fainter apparent bright-
ness (Kp=15.4 and 14.1, respectively). Four additional Ke-
pler RR Lyrae stars show weak signs of the period-doubling
phenomenon. All these objects show the enigmatic Blazhko
effect, i.e., amplitude and phase modulation of the regu-
lar RR Lyrae pulsation. For the recent Kepler findings with
respect to RR Lyr itself and an overview of the Blazhko be-
havior of Kepler RR Lyrae stars we refer to Kolenberg et al.
(2010b) and Benk˝ o et al. (2010), respectively.
Since resonances are known to play much less of a role
in RR Lyrae stars than in Cepheids, it is natural to ask
whether occurrence of any type of resonance between radial
modes can cause this behavior. Serendipitously, we encoun-
tered RR Lyrae models showing the PD bifurcation and sub-
sequently we applied them to the Kepler RR Lyrae stars. We
were able to demonstrate that in RR Lyrae stars the physical
origin of this instability is a 9:2 resonance between the fun-
damental mode and a high-order radial overtone. This lat-
ter mode is called a strange mode (Buchler & Koll´ ath 2001),
because it has no adiabatic counterpart and the energy as-
sociated with its pulsation is confined to the outer zones of
the star. The goal of this work is to present the first Kepler
RR Lyrae period doubling results and our first successful
modeling efforts.
The outline of this paper is as follows. In Sec. 2. we
describe Kepler observations we use and devote special em-
(1990,
reported
1991)
period
and
doubling
phasis to reduction and proper handling of Kepler photom-
etry. In the next section the observed properties of the pe-
riod doubling are presented. Next we turn to hydrodynam-
ical models that successfully reproduce the newly discov-
ered phenomenon in Sec. 4. Finally, the implications of using
this transient phenomenon to gain insight to the enigmatic
Blazhko effect are explored.
2 OBSERVATIONS
Kepler was designed to detect transits of terrestrial planets
on Earth-like orbits around solar-like stars. This requires the
observation of ∼ 105main sequence stars continuously for
several years with great accuracy. Kepler was launched on
2009 March 6, and observes a 105 square degree area of the
sky in constellations Cygnus and Lyra, a few degrees above
the galactic plane. After a short commissioning phase, the
scientific observations started on May 12. In order to ensure
optimal solar irradiation of the solar arrays, a 90degree roll
of the telescope is performed at the end of each quarter.
The first roll lasted only for 33.5days (Q1). The second roll
was the first complete one (Q2). In this work we use both
Q1 and Q2 when available, i.e., 127days quasi-continuous
observations.
The Kepler magnitude system (Kp) refers to the
wide pass band (430 − 900nm) transmission of the
telescope and detector system. Both long-cadence (LC,
29.4min, Jenkins et al. 2010), and short-cadence (SC, 58.9s,
Gilliland et al. 2010b) observations are based on the same
6-s integrations which are summed to form the LC and SC
data onboard. In this work we used only long-cadence data.
The saturation limit is between Kp ≃ 11 − 12mag depend-
ing on the particular chip on which the star is observed;
brighter than this, accurate photometry can be performed
up to Kp ≃ 7mag with judiciously designed apertures.
The Kepler Asteroseismic Science Consortium (KASC)
was set up to exploit the potential of Kepler in solar-like
oscillations as well as all types of pulsations. KASC Working
Group#13 is dedicated to the investigation of RR Lyrae
stars. Out of 29 RRab stars that were observed by Kepler
14 were found to be Blazhko stars. Small gaps are seen in
their light curves (Fig.3). These are due to unplanned safe
mode and loss of fine point events as well as regular data
downlink periods. Excluding these cadences, the Q1 data
segment contains 1626 useful data points, while Q2 contains
4097 points.
Our trend and jump filtering algorithm was tested on
the Kepler data. We found that no detrending was neces-
sary for our targets. Jump corrections were also considered
unnecessary for our purposes. We noticed that Q1 and Q2
mean brightness and amplitudes of a given pulsating star
may differ. For our fainter targets only the mean brightness
had to be adjusted between different rolls. In the case of
RR Lyr, however a more thorough analysis and calibration
was needed.
2.1Accurate photometry of bright Kepler targets
As we mentioned before, Kepler CCDs saturate between
Kp ≃ 11 and 12mag, but the saturated flux is conserved
to a very high degree, spilling in the column direction. This

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Period doubling in Kepler RR Lyrae stars3
Table 1. Main properties of the observed Kepler Blazhko RR Lyrae stars showing the period doubling effect. Errors are given in
parenthesis implying variation in the last digit only. The uncertainty of the Blazhko period is estimated to be 0.d3.
KIC ID GCVS nameR.A.
(J2000)
Dec.
(J2000)
Kp
[mag]
Puls. period
[days]
A1
[mag]
Blazhko period
[days]
Runs
7198959
4484128
7505345
RR Lyr
V808 Cyg
V355 Lyr
19 25 27.91
19 45 39.02
18 53 25.90
+42 47 03.73
+39 30 53.42
+43 09 16.45
7.862
15.363
14.080
0.5669685(8)
0.5478721(8)
0.4736958(10)
0.158(2)
0.299(3)
0.374(3)
39.6
90.2
31.3
Q1,Q2
Q1,Q2
Q2
5.50265.5027 5.50285.5029 5.503 5.50315.50325.50335.5034
x 10
4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
x 10
10
MJD
flux (electrons)
Figure 1. A comparison between the captured flux from RR Lyr
(blue dashed) and the corrected flux (red) during Q2, with ±1σ
error bars. Note the alternating high/low amplitudes and depths
at minimum and maximum light.
5.50265.50275.50285.50295.5035.50315.50325.50335.5034
x 10
4
2.5
3
3.5
4
4.5
5
5.5
x 10
9
MJD
flux in column (electrons)
Figure 2. The flux summed along the left-adjacent (blue dashed)
and right-adjacent (red) columns to the central column. The in-
cident stellar flux in these adjacent columns was fully captured.
allows Kepler to perform high-precision photometry on sat-
urated targets like RR Lyr as long as a sufficient number of
pixels is captured in the column direction. In Q1 and Q2, a
fraction of flux from RR Lyr fell outside the Kepler aperture
(set of downlinked pixels), so extra care is required to assure
that the period doubling phenomenon is not due to loss of
flux in some cadences. We developed an approach that esti-
mates a correction of the RR Lyr flux. Part of the original
and corrected light curve is shown in Fig. 1. The details of
this method are described in Kolenberg et al. (2010b).
Here we only note that in Q1 the flux corrections in case
of the RR Lyr were less than 5%, while in Q2 the necessary
flux corrections were larger, ranging from 15% to 35%. The
uncertainty in the total corrected flux is about 0.25%. This
is to be compared with the flux uncertainty of 8×10−6in ca-
dences where the correction was not applied because the flux
was completely captured. For comparison the uncertainty of
the (uncorrected) flux for the two fainter stars showing the
period doubling varies between 3 − 7 × 10−4for V808 Cyg
and 1.5 − 3.1 × 10−4for V355 Lyr.
Because these corrections are estimates, we provide the
following evidence that the period doubling phenomenon in
Q2 RR Lyr data is not due to loss of flux:
• Similar alternating cycles are seen in the flux time se-
ries of individual pixels, as well as the sum of pixels along
columns with significant flux (other than the central col-
umn) from RR Lyr (see Fig. 2). In individual pixels, the
period doubling signal is much larger than the pixel-level
noise, which is dominated by pointing jitter and shot noise.
• Saturated pixels induce a faint video crosstalk signal on
other CCD channels. At its maximum, crosstalk from the
saturated flux from RR Lyr impinged on a faint observed
target, and this crosstalk signal is consistent with the recon-
structed light curve including period doubling.
• Kepler light curves are created using pixels in a pho-
tometrically optimal aperture that maximizes the signal to
noise ratio for the target (Bryson et al. 2010). Kepler down-
links a superset of these pixels (Haas et al. 2010) including
at least a 1-pixel halo around the photometrically optimal
aperture. In the case of Q1 and Q2 observations of RR Lyr,
significantly more pixels were downlinked. Flux light curves
created using all downlinked pixels exhibit essentially the
same period doubling phenomenon as the light curve gener-
ated from the photometrically optimal pixels. This demon-
strates that period doubling is not due to loss of flux from
the (smaller) optimal aperture other than the central satu-
rated column.
• Perhaps the strongest evidence that period doubling
is not due to loss of flux is that the period doubling phe-
nomenon was seen in Q1 when the flux from RR Lyr was
completely captured.
• The fact that three stars were found unambiguously
showing the PD effect further helps in ruling out external
influences. The bright RR Lyr and the two much fainter
stars (V808 Cyg and V355 Lyr) demonstrate that the same
effect is operational at two different parts of the dynami-
cal range of the CCDs. In addition, all three stars fell on
different CCD-modules and the modules were rotated be-

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4R. Szab´ o, Z. Koll´ ath, L. Moln´ ar, et al.
7.20
7.40
7.60
7.80
8.00
Kp [mag]
RR Lyr
0.0
54960
5.0
10.0
15.0
20.0
25.0
55100550805506055040 5502055000 54980
An/2 [mmag]
HJD-2400000
1/2 f0
3/2 f0
5/2 f0
7/2 f0
9/2 f0
Figure 3. Upper panel: Q1+Q2 light curve of RR Lyr. Note that the individual pulsational cycles are hardly discernible, while the long
period (39.6d) Blazhko modulation clearly stands out. The three dashed boxes are blown-up in Fig 4. Bottom panel: amplitudes of the
half-integer frequencies.
8.0
7.8
7.6
7.4
54994 54990
Kp [mag]

5503455030
HJD - 2400000
5507055074

Figure 4. Seven-day segments of the Kepler light curve of RR Lyr showing different degree of period doubling effect at the same Blazhko
phase. The maxima and the minima were fitted with a 9th order polynomial and the small horizontal bars drawn through the extrema
are plotted to guide the eye.
tween Q1 and Q2, so the PD effect is independent of the
CCD-modules.
Based on the analysis described in this section, we have
high confidence that the period doubling phenomenon is not
due to instrumental effects.
3THE PERIOD DOUBLING PHENOMENON
3.1Alternating extrema and half-integer
frequencies
Period doubling is found in three of the Blazhko stars
in the Kepler field: RR Lyr (KIC7198959), V808 Cyg
(KIC4484128) and V355 Lyr (KIC7505345). Some of their
properties can be found in Table. 2. The pulsational period,
the Blazhko period and the amplitude of the first Fourier-
component (A1) were derived from the available Kepler ob-
servations. These numbers will be refined with more Kepler
data.
The upper panels of Figs 3 and 5 show the Q1+Q2
light curves for RR Lyr and V808 Cyg in the Kp band. The
individual maxima and minima were fitted with a 9th order
polynomial to test the effect of the 29.4-min sampling which
may undersample the rapidly changing light curve around
the maxima. We find no significant problem arising from
the long-cadence sampling. Certain parts of the light curves
are marked with dotted line rectangles and are magnified to
discern the alternating maxima and minima in Figs 4 and
6. The fitted maxima and minima are plotted as horizontal
bars to guide the eye in these figures. In the case of RR Lyr
a difference of 0.m1 is seen in the brightness of subsequent
maxima. This amounts to a few hundredths of magnitude in
case of V808 Cyg.
The alternating maxima and minima in conjunction
with the half-integer frequencies (HIFs, i.e., k/2 · f0, where
k = 1,3,5,...) in the frequency spectrum are typical signs
of the period doubling bifurcation. For the frequency anal-
ysis we used SigSpec (Reegen 2007). Where available, Q1
and Q2 data sets were merged. When the spectral signifi-
cance reached the conservative value 5 the procedure was
stopped, although the traces of HIFs can be followed up to
the Nyquist-frequency (24.5c/d). The results were checked
by Period04 (Lenz & Breger 2005). Only minor differences
were found, mainly in the phase values.
In our third case, V355 Lyr, the PD effect is undoubt-
edly present, but is rather weak (Fig. 7). The maximum

Page 5

Period doubling in Kepler RR Lyrae stars5
14.6
14.8
15.0
15.2
15.4
15.6
15.8
Kp [mag]
V808 Cyg
0.0
54960
5.0
10.0
15.0
20.0
25.0
5510055080 550605504055020 55000 54980
An/2 [mmag]
HJD-2400000
1/2 f0
3/2 f0
5/2 f0
7/2 f0
9/2 f0
Figure 5. Upper panel: Q1+Q2 light curve of V808 Cyg. The three dashed boxes are blown-up in Fig 6. Bottom panel: amplitudes of
the half-integer frequencies.
14.8
15.0
15.2
15.4
15.6
15.8
54986 54990

54994
Kp [mag]
550405504455048
HJD - 2400000
5507455078

55082
Figure 6. Twelve-day segments of the Kepler light curve of V808 Cyg showing the period doubling effect at different Blazhko phases.
amplitude of the 3/2f0 frequency is 5 mmag, while it is
25 mmag for the two other targets. This translates to a few
hundredths of a magnitude difference in consecutive max-
ima or minima. Interestingly, the amplitude modulation is
also small for this star.
Four other Blazhko RR Lyrae stars in the Ke-
fieldareseentopossibly
fect: V2178 Cyg (KIC3864443), V354 Lyr (KIC6183128),
V445 Lyr (KIC6186029) and V360 Lyr (KIC9697825). In
their frequency spectra peaks were found close to the pre-
dicted half-integer frequencies. However, our criteria for the
detection of the PD effect were the clear sign of alternating
height of the pulsation cycles as well as the simultaneous
presence of a large number of HIFs (preferably more than
eight). If any of these two requirements were not met by a
star, we consider it as a possible PD object only. We note
that some of these four stars in this category show additional
frequencies making their frequency spectrum more complex.
For more details on these stars we refer to Benk˝ o et al.
(2010).
pler
exhibit thePDef-
From now on we turn to our three stars that show se-
curely detected PD phenomenon. We plotted the averaged
amplitudes of the HIFs of these stars in Fig. 8 taken from
their frequency spectra. It is interesting to note that in all
three cases the 3/2f0 frequency has the highest amplitude
among the half-integer frequency peaks, next comes 5/2f0
and 1/2f0. This appears to be a general feature of the PD
phenomenon in Blazhko stars. Around the fifth half-integer
frequency, (i.e., k = 9) a pronounced bump is seen in the
amplitude distribution. The origin of this bump is explained
in detail in Sec. 4. The amplitude of the higher-order half-
integer peaks are decreasing more or less steadily with the
order number k.
3.2The transient nature of the period doubling
After discussing the time-averaged properties of the HIFs we
now turn to investigate their temporal behavior. The lower
panels of Figs 3 and 5 for RR Lyr and V808 Cyg respectively,
show the temporal behavior of the amplitude of the most
prominent half-integer frequencies in the frequency spec-
tra. These were computed using the analytic signal method
(Koll´ ath et al. 2002), a powerful method developed to follow
time-dependent signals. We note here that the method is su-
perior compared to other time-dependent Fourier-methods,
but has a drawback: in the presence of large gaps the proce-
dure does not yield reliable results, therefore we had to cut
the neighborhood of the missing data. We found a 0.25c/d
bandwidth to give the most stable results, and a ∼ 2.5d
long data segment is lost in each side of a gap. The rela-
tively broad bandwidth means that sometimes more than
one frequency peak is contained in the computed interval

Page 6

6R. Szab´ o, Z. Koll´ ath, L. Moln´ ar, et al.
but the insensitivity to noise compensates for this disadvan-
tage. We tested that the temporal behavior of the HIFs is
not flawed by the chosen bandwidth.
It is obvious from Figs 3 and 5 that the intensity of
the period doubling phenomenon is changing with time. For
RR Lyr it has maximum strength on the ascending branch
of the Blazhko envelope on the first two rising branches, and
is much less visible during the third ascending branch of the
Blazhko modulation. The difference in brightness between
consecutive maxima reaches as high as 0.1mag when PD is
strongest. The visible alternating extrema can be seen where
the amplitude of the HIFs is high, and no significant alter-
nation is found where the amplitude is low. This is true for
all our target stars throughout the whole light curve in each
case. In RR Lyr the amplitudes of the HIFs never vanish, in
other words the PD effect is always present. Fig. 9 demon-
strates the difference of the strength of the HIFs showing the
discrete Fourier transforms of the second and third marked
segments of the RR Lyr light curve as shown in Fig. 3. We
prewhitened with the main pulsation frequency and its har-
monics (but not with the Blazhko side-peaks) for better vis-
ibility of the HIFs. It is discernible that the half-integer fre-
quencies are present in both data segments, with a factor
of three difference in the amplitudes. We can conclude that
the PD effect is transient and varies with the Blazhko cycle.
In the case of V808 Cyg the PD effect is practically
seen throughout the whole 133-d observational period, albeit
with outstanding maxima of the HIFs during the ascending
branch of the Blazhko envelope, close to the Blazhko maxi-
mum and during the descending branch. The minimum level
of the half-integer frequencies is not as low as for RR Lyr,
but the maximum height is similar. Again, there is a clear
connection of the PD effect to the Blazhko modulation, very
similar to the case of RR Lyr, but with enhanced intensity
and an additional maximum height during the descending
branch of the Blazhko envelope.
One can see numerous peaks in the vicinity of the half-
integer frequencies (Fig. 10c). In addition, we noticed that
the highest frequency peak has a frequency ratio to f0 that
is significantly different from 2/3, namely 0.662. This is the
combined effect of the Blazhko modulation and the temporal
onset and disappearance of the HIFs. To test this hypothesis
we performed the following check.
We generated an artificial light curve sampled at the
original data points. We took f0 and its 16 harmonics of
RR Lyr, their phases and amplitudes and modulated the
amplitude and the phase with a Fourier-sum of two terms
and five terms, respectively. The resulting light curve is very
similar to the observed one, but we note that the purpose of
this simulation was to explain the frequency spectrum in the
vicinity of the HIFs and not the reproduction of the light
curve.
Then we added the 1/2f0, 3/2f0, 5/2f0 etc. frequency
series with small amplitude. The same modulation is applied
to these periodic signals, as well. We used f0 = 1.762989 c/d
in the simulation and applied MuFrAn for the frequency
analysis (Koll´ ath 1990). It resulted in f′
prewhitened the light curve with f′
The resulting spectrum between f0 and 2f0 is shown in the
upper panel of Fig. 10, where additional small peaks appear
around 3/2f0.
The middle panel shows the result of a similar pro-
0= 1.763059 c/d. We
0and its 16 harmonics.
13.4
13.6
13.7
14.0
14.2
14.4
550305503255034
Kp [mag]
HJD-2400000
Figure 7. Six-day segment of the V355 Lyr light curve showing
small period doubling effect, i.e., alternating maxima and min-
ima.
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
252015105
amplitude [mag]
order k
RR Lyr
V808 Cyg
V355 Lyr
Figure 8. Amplitudes of the half integer frequencies as a function
of the order k, where k denotes the k/2 · f0 frequency. Note the
significant bump seen around k = 9 as a sign of the 9:2 resonance
with the 9th (strange) overtone for V808 Cyg and the change of
slope for the other stars.
cedure, but here the period-doubling (i.e., the HIFs) was
switched on and off periodically. Specifically, the amplitudes
of the HIFs were varied as a = sin(fm∗ t + φ) if a > 0 and
a = 0 otherwise, i.e., HIFs are present only when the func-
tion is positive. fm denotes the modulation frequency. One
can find at least 5 distinct peaks in the synthetic spectrum,
the highest two of them are of similar amplitude. The fre-
quencies of these peaks are: f′= 2.6432455 c/d and f′′=
2.6701858 c/d, their frequency ratios are: f′/f′
f′′/f′
Our simulation demonstrated convincingly that the
large number of frequency peaks and the frequency ratio
close to but not equal to 2/3 are expected consequences of
the modulated light curve, and we are dealing with a genuine
period doubling effect.
Summarizing our findings concerning the PD effect in
three Kepler Blazhko RRab stars we conclude that this ef-
fect occurs in some but not all modulated RR Lyrae stars.
For a given star its presence may be continuous, but there
are specific phases of the Blazhko cycle, where it gets much
stronger (up to five times in amplitude). This is not a strict
0= 0.667 and
0= 0.660, respectively.

Page 7

Period doubling in Kepler RR Lyrae stars7
-20
-10
0
10
20
30
0246810
amplitude [mmag]
frequency [c/d]
2nd segment
3rd segment
1/2f0
3/2f0
5/2f0
7/2f0
9/2f0
11/2f0
Figure 9. Fourier spectrum of the second (upper panel) and
third (lower panel) segments of the Kepler light curve of RR Lyr
as shown on Fig.3. (between MJD 55027-35 and 55068-76, respec-
tively). We prewhitened by the main pulsational frequency (f0)
and its harmonics for clarity. The side-peak structure around f0
and its harmonics are still visible as well as the variable strength
of the half-integer frequencies.
rule, however. Out of three rising branches of the Blazhko
modulation (i.e., envelope) in RR Lyr we detected a strong
presence of PD in the first two cases, and much weaker ap-
pearance during the third one. The overall dominance of the
period doubling may also vary, for RR Lyr and V808 Cyg it
is relatively strong, while for V355 Lyr it is much weaker.
3.3Detection limit in the Kepler sample
Upper limits for the averaged amplitudes of half-integer fre-
quencies were established for all the Kepler RR Lyrae stars
where we did not find PD effect. Frequency spectra were
computed for the 14 Blazhko and 15 non-Blazhko Kepler
RR Lyrae stars) using SigSpec and prewhitened successively
with the highest frequency peaks. These were the dominant
pulsational mode (fundamental in each cases), its harmon-
ics up to the Nyquist-frequency and modulation side peaks
in case of Blazhko stars. The procedure was stopped when
the amplitudes of the remaining peaks reached a spectral
significance of 5.0. As the limit for the HIF amplitudes we
accept the amplitude corresponding to this spectral signifi-
cance limit. We emphasize that this choice is conservative,
because inspection of the location of the HIFs showed that
there were no frequency peaks up to an amplitude two to
three times lower than our adopted limit of 5.0.
Two factors affect our detection limit, one is the bright-
ness of the star and the other the complexity of the frequency
spectrum (see Benk˝ o et al. 2010). We note in passing that
the apparent magnitudes of the Kepler RR Lyrae sample are
in the range of Kp ≃ 11 − 17, the notable exception being
RR Lyr, which is much brighter. Another source of error may
be the lack of barycentric correction to the (shorter) Q1 time
series. While this is correction is important, we argue that it
has no effect on our detection limit. First because it does not
effect the alternating maxima and minima. Second, although
it may cause some systematics in the frequency spectrum,
the basic structure of the HIFs is well understood, as we
demonstrated in the previous subsection.
Frequency [c/d]
Amplitude [KP]
a)
0
.005
.01
.015
b)
0
.002
.004
.006
c)
1.6 1.822.2 2.42.6 2.833.2 3.43.6
0
.002
.004
.006
Figure 10. a) Frequency spectrum between f0and 2f0of a syn-
thetic RR Lyr light curve showing the 3/2f0frequency peak. The
simulation was performed by keeping only f0, its harmonics and
the k/2f0 frequencies (k = 1,3,5,...), and the same modulation
is applied for all these frequencies. b) The same as on the upper
panel, but the HIFs are switched on and off resulting in a bunch of
additional frequencies. c) The frequency spectrum of Q2 Kepler
RR Lyr data plotted between f0 and 2f0.
As we mentioned in Sec. 3.1., we found four additional
stars showing some HIFs besides RR Lyr, V808 Cyg and
V355 Lyr. For the remaining Blazhko stars not showing the
PD effect we find that the upper limit for the HIF ampli-
tudes is between 0.2−2.0 mmag. For RR Lyrae stars without
modulation we generally find a smaller upper limit. The de-
tection limit for these objects is between 0.1 − 1.0 mmag,
with more stars lying closer to the 0.1 mmag border line.
We have not found period doubling bifurcation in the
ultra-high precision light curve of any of the non-Blazhko
stars, in spite of the fact that it would be much easier to dis-
cern alternating extrema, as well as HIFs in their frequency
spectra. In conclusion, these findings strongly suggest that
the PD bifurcation is related to the Blazhko effect.
4HYDRODYNAMICAL SIMULATIONS
Hydrodynamical modeling has proved that low order res-
onance plays an important role in the oscillations of
classical pulsating stars. The Hertzsprung progression of
bump Cepheids is traced to the P0/P2=2 resonance (e.g.
Buchler et al. 1990). The sharp features in the Fourier co-
efficients of s-Cepheids are traced to the P1/P4 = 2 reso-

Page 8

8R. Szab´ o, Z. Koll´ ath, L. Moln´ ar, et al.
20
30
40
50
60
70
80
0 5 10 15 20 25
L/L0
time [d]
Figure 11. Results of a hydro run showing the onset of the period doubling phenomenon.
nance, despite the very large damping of the fourth overtone
(Feuchtinger et al. 2000). In the case of BL Her and Cepheid
type pulsations it was demonstrated by Moskalik & Buchler
(1990) that 3:2 resonance of the fundamental mode and the
first overtone is responsible for period doubling bifurcation
in hydrodynamical models.
However, only the low order modes up to the 4th over-
tone were considered when resonances were discussed, and
the remaining modes were assumed to have no influence
on the asymptotic behavior of the models since they are
strongly damped.
In RR Lyrae stars, effects due to the above mentioned
resonances are not expected because of the different pe-
riod ratios. However, in some of the RR Lyrae model se-
quences period doubling bifurcation was detected, indicat-
ing that indeed there exists some mechanism that is able to
destabilize the fundamental mode pulsations. To pinpoint
the mechanism behind the period doubling bifurcation we
have performed a systematic survey of RR Lyrae model
sequences. The details of these calculations are presented
elsewhere (Koll´ ath, Moln´ ar & Szab´ o 2010), here we present
only the results relevant to this paper. For our hydrody-
namical calculations we used our standard turbulent con-
vective stellar pulsation hydro-code (Florida-Budapest code,
see Koll´ ath et al. 2002, Eqs. 1–13.) The main model param-
eters are M = 0.578M⊙, L = 38.45L⊙, Teff = 6500K, and
metallicity: Z=0.0001.
Standard integration of the models with PD behavior
evolve to the bifurcated solution and thus the fundamental
mode limit cycle cannot be calculated in this way. However,
the relaxation method (see Kov´ acs & Buchler 1993) makes
it possible to iterate to the limit cycle solution and determine
its stability properties. These calculations indicate that the
fundamental mode limit cycle is unstable for a wide range
of the model parameters. The large value of one of the Flo-
quet exponents (λk ≈ 0.5) indicates that perturbations to
the limit cycle grow on a time scale of a few periods. Time
integration of the model initiated from the limit cycle so-
lution clearly demonstrates the short timescale of the tran-
sition from limit cycle to period-2 solution. The luminosity
variation during this transition is displayed in Fig. 11. The
perturbation to the limit cycle was defined as a 1% increase
of eddy viscosity in the model. We have to note, however,
that with no direct perturbation the model also evolves to
the PD solution on a slightly longer timescale due to the
numerical noise of the computations.
The numerical integration of the model clearly demon-
strates the PD bifurcation in our RR Lyrae model. However,
it does not provide a direct clue on the destabilizing mecha-
nism of the fundamental mode. The PD bifurcation can oc-
cur through the destabilization of either a thermal mode or a
vibrational mode. The second case was thoroughly described
by Moskalik & Buchler (1990), showing that a half-integer
resonance provides the mechanism in period doubling bifur-
cation in hydrodynamical models. The Floquet coefficient
that gives the instability of the limit cycle is real (the Flo-
quet phase, φk = π). It indicates that the coupling to a vi-
brational mode is in effect through a half integer resonance,
but it does not rule out the possibility of a thermal mode
behind the PD behavior. Resonance is not expected in the
low order modes, however, the linear stability analysis of the
model sequences shows that some of the higher (8th-10th)
overtones are unstable for some of the temperatures. This
behavior of the linear models indicates that a strange mode
coexists with the normal vibrational modes, suggesting that
a resonance with this strange mode can be responsible for
the destabilization of the fundamental mode.
In Fig. 12 the period ratios P0/Pk are displayed for the
high-order modes. The period ratio curves show signatures
of avoided crossings proving the existence of a strange mode
(Buchler & Koll´ ath 2001). Interestingly, it clearly shows a
half integer resonance with P0 : Pk = 9 : 2 in a wide tem-
perature range, which was not expected to provide such a
strong influence on fundamental mode pulsation. A thor-
ough nonlinear hydrodynamical survey of model sequences
demonstrates that indeed this resonance provides the desta-
bilization of the fundamental mode and it causes the period
doubling bifurcation. Details of these calculations are pre-
sented in a parallel paper (Koll´ ath, Moln´ ar & Szab´ o 2010).
The PD behavior strongly depends on the parameters
of turbulent convection. Using e.g., the eddy viscosity as
a control parameter, the bifurcation from limit cycle to
period doubling is obtained. Similarly the convective effi-
ciency can play an important role as a control parameter.
If one assumes that during the Blazhko cycle the turbu-
lent/convective structure of the star varies (Stothers 2006),
it can result in the repeated occurrence of PD similarly to
the observations.

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Period doubling in Kepler RR Lyrae stars9
3.5
4
4.5
5
5.5
6
6.5
7
5500 6000 6500 7000 7500
P0/Pk
Teff [K]
o7
o8
o9
o10
o11
Figure 12. Linear period ratios of high-order radial overtones
and the fundamental mode as a function of effective temperature.
The 9:2 resonance is shown by a horizontal line.
5DISCUSSION
The high precision and continuity of the Kepler space tele-
scope has enabled us to discover a new phenomenon in
Blazhko RR Lyrae stars, namely period doubling. Three
stars were found to show this type of behavior unam-
biguously. One of them is the brightest representative of
its class RR Lyr (KIC7198959) and the other two are
much fainter modulated RRab variables, namely V808 Cyg
(KIC4484128) and V355 Lyr (KIC7505345). In addition,
four other Blazhko RR Lyrae stars in the Kepler field may
show signs of this new type of instability.
Period doubling manifests itself as alternating maxima
and minima of pulsational cycles, sometimes even the shape
of the light curve is alternating. As a consequence, the
Fourier spectrum contains half-integer frequencies (HIFs),
i.e., frequency peaks midway between harmonics of the main
pulsational frequency.
Interestingly enough, the intensity of the PD effect is
time-dependent. In RR Lyr it is most prominent during the
ascending branch of the modulation in two Blazhko cycles,
while it is practically missing during the third ascending
branch. In V808 Cyg the PD effect is seen throughout the
whole 133-d long data set, albeit with outstanding maxima
of the half-integer frequencies during the ascending branch
close to the Blazhko maximum and during the descending
branch. In V355 Lyr the PD effect is present, but it is rather
weak, the maximum amplitude of the 3/2f0 frequency is
five times less than in the case of the two other targets.
Similarly, the amplitude modulation is also rather small for
V355 Lyr. This may hint at an intimate connection between
the strength and of the Blazhko modulation and the period-
doubling effect. Also, the fact that no PD effect was found in
non-Blazhko Kepler RRab stars strongly suggests that this
effect is connected to the Blazhko effect.
The structure of the HIFs in the spectrum is found
to be rather complex. We successfully demonstrated that
the bunch of appearing frequency peaks in the vicinity of
the expected HIFs is due to the varying pulsational period
throughout the modulation cycle, as well as the transient
nature of the period doubling phenomenon.
Although deviations from regular single-periodic pulsa-
tion, i.e., cycle-to-cycle variations in RR Lyrae radial veloc-
ity curves (Chadid 2000) and irregularities in photometric
observations (Jurcsik et al. 2008) had already been detected
from the ground, one might ask why the PD effect was not
discovered despite the fact that the difference of the subse-
quent maxima during the PD episode may well be observable
from the ground with accurate CCD photometry. Firstly,
3/2f0 (the highest half-integer frequency in all three PD
Blazhko stars) grows to 26mmag in the Fourier-spectrum
of RR Lyr when it shows maximum power. It may be visi-
ble only in well-sampled (essentially continuous) light curves
which is very hard to obtain. Before MOST, CoRoT and Ke-
pler no continuous RR Lyrae light curve was available. The
compact, dedicated single-site observations (Jurcsik et al.
2005, 2008) and the limited multisite campaigns that have
been organized for RR Lyrae stars (Kolenberg et al. 2006,
2009) did not yield the required coverage and accuracy to
be able to detect the PD phenomenon. Secondly, pulsation
periods close to 0.d5 (typical for RR Lyrae stars) may ham-
per the detection, as from the ground one can follow only
even or odd cycles every night and the duration of the max-
imum strength of the PD phenomenon as we see in Kepler
data is not long (typically 8–10d). Finally, period doubling
itself is a transient phenomenon, not seen in every Blazhko
cycles. We estimate that the maximum PD strength phase
(i.e., possibly observable from the ground) lasts from 10%
(RR Lyr) to 22% (V808 Cyg) of the currently available time
span covered by Kepler observations. That’s where the con-
tinuity and longevity of Kepler observations have unbeat-
able advantage. In addition, strong PD occurs only in three
stars out of 14 RR Lyrae exhibiting the Blazhko modulation,
while four additional modulated RR Lyrae stars show much
weaker evidence for PD-like behavior. We conclude that it is
not surprising that the transient PD effect remained unno-
ticed in decades-long ground-based RR Lyrae observations.
The period-doubling bifurcation was reproduced suc-
cessfully with the Florida-Budapest hydrocode which ac-
counts for turbulent convection. We emphasize that not only
the period doubling effect occurs naturally in our hydrody-
namical models, but the time scale of the onset and fade-out
of the PD effect is excellently reproduced, as well.
Our models of RR Lyrae stars demonstrated that pe-
riod doubling is possible in these stars due to a 9:2 reso-
nance of the fundamental and a high order (9th overtone)
mode. It was not expected that such a high order mode plays
an important role in fundamental mode pulsations. How-
ever, it was found that this interacting mode is a strange
mode, with a non-normal damping rate and eigenfunction.
Normal high order pulsation modes, with this extreme pe-
riod ratio (P0 : Pk = 9 : 2), are not able to destabilize the
fundamental mode limit cycle and to induce a period dou-
bling bifurcation. Thus the observed period doubling char-
acteristic in the Kepler RR Lyrae stars provides a strong
indirect evidence for the existence of strange modes in ra-
dial stellar pulsation, a phenomenon predicted theoretically
by Buchler et al. (1997). The significant interaction of the
strange mode to the fundamental mode pulsation also sug-
gests that strange modes can play an important role in
other phenomena, like three-mode resonances (e.g., among
the fundamental, 1st/2nd, and the strange mode); and per-
haps it has an effect in shaping the Blazhko effect as well. In
addition, nonradial modes may also be involved in this com-

Page 10

10R. Szab´ o, Z. Koll´ ath, L. Moln´ ar, et al.
plex dynamical interplay through resonant or non resonant
interactions, as demonstrated by recent Kepler findings in
Benk˝ o et al. (2010).
We note that we found the strongest phase (or equiv-
alently period) modulation in the cases of RR Lyr and
V808 Cyg among Kepler Blazhko stars (Benk˝ o et al. 2010),
and these stars show the strongest PD effect. If we as-
sume that during the Blazhko cycle the turbulent/convective
structure of the star varies as suggested by Stothers (2006),
it seems natural that in certain Blazhko phases (i.e., when
the physical conditions are favorable) the PD effect appears,
because in our models the PD behavior strongly depends
on the parameters of turbulent convection. This sensitivity,
together with the narrow parameter range where the neces-
sary (P0 : Pk = 9 : 2) resonance is at work make this new-
found phenomenon a precious tool for studying the myste-
rious Blazhko effect. The understanding of the differences
between PD and non-PD Blazhko RR Lyrae stars, as well
as the strong and feeble PD phases of a given star may pro-
vide the long-sought insight into the Blazhko mechanism,
offering a sensitive way to constrain our models.
With the release of additional Kepler data it will be
possible to further study the PD behavior and learn more
about its temporal and transient nature.
ACKNOWLEDGMENTS
Funding for this Discovery mission is provided by NASA’s
Science Mission Directorate. This project has been sup-
ported by the National Office for Research and Technology
through the Hungarian Space Office Grant No. URK09350
and the ‘Lend¨ ulet’ program of the Hungarian Academy of
Sciences. KK acknowledges the support of Austrian FWF
projects T359 and P19962. The authors gratefully acknowl-
edge the entire Kepler team, whose outstanding efforts have
made these results possible.
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