Astron. Astrophys. 339, 440–448 (1998) ASTRONOMYAND

ASTROPHYSICS

Studies on amplitude modulated RR Lyrae stars

II. RS Bootis

A. Nagy

Konkoly Observatory, Budapest XII, P.O. Box 67, H-1525 Hungary (nagya@buda.konkoly.hu)

Received 30 June 1998 / Accepted 28 July 1998

Abstract. All photographic and photoelectric observations ofRS Boo since 1938 have been analysed in order to obtainan accurate mathematical description of the light variation. AFourier sum with frequenciesf0, fm, kf0, kf0 ± fm, wherek = 1, 2, . . . , 11; f0 = 2.6501491 d−1; fm = 0.001886 d−1

and a linear period change of thef0 and thekf0 componentswith β = 7.0 × 10−11 give a fit with a standard deviation ofσ = 0.064 mag. Considering the various quality and the longspan of the data, this level of accuracy can be regarded as agood one. In contrast with some reports about the existenceof a secondary modulation we have not found any sign of thiscomponent in the prewhitened spectrum.

Key words: stars: individual: RS Boo – stars: variables: RRLyrae – stars: oscillations

1. Introduction

Most of the fundamental mode RR Lyrae stars pulsate strictlyperiodically, but about a third of them show periodic orquasiperiodic amplitude and phase variations on a longertimescale (Szeidl 1988). This phenomenon, commonly calledasBlazhko effect, was first observed by Blazhko (1907) in thecase of the RRab-type star RW Dra. The traditional O – Canal-ysis requires data only in specific phases of the pulsation, there-fore, most of the observations were made at or near maximumbrightness and on the ascending branch. Fourier analysis of atime series with this type of quasi-periodic sampling is a diffi-cult problem, hence there are not many works concerning thisquestion (Almar 1961; Borkowski 1980; Goranskij 1989; Smithet al. 1994; Kovacs 1995). It would be very important to havean accurate mathematical description of the light variation ofthese stars in order to establish a more exact basis for studyingthe physics of the amplitude and phase modulations.

The variability of RS Boo was first observed more than ahundred years ago by Fleming (Duner et al. 1907). The ampli-tude and phase modulations were first pointed out by Oosterhoff(1946), but one can find some interesting remarks concerningthe height of the maxima already in the paper of Lause (1931).During this century, many observations have been made, but,unil now, no Fourier analysis has been performed. RS Boo is avery intresting star because it has an extremely long modulationperiod — it is by far the longest among the Blazhko variables.

Therefore, it is important to describe its light variation and ob-tain a mathematical model which can be compared with theones obtained for the other Blazhko stars of shorter modulationperiod.

2. The data

The analysis presented in this paper is based on two main datasets. The first one is the well-covered photographic observa-tional series of Oosterhoff (1946) obtained between 1938 and1944. This set of observations spans seven years without anylong gaps and covers all phase values. The second one is a pho-toelectric series. Most of the observations were made by Kanyo(1986) between 1959 and 1976. These data cover primarily theascending branch and the maximum. There is a twelve-year-long gap in this data set, which is partially covered by the ob-servations of Fitch et al. (1966) and Ste¸pien (1972). These setscover the minimum, the ascending branch and the maximum.The data of Spinrad (1959) — obtained just before Kanyo’s firstset of data — cover all phase values. There is another short seg-ment of observation covering all phase values by Jones et al.(1988). This set follows Kanyo’s data with an eight-year-longgap. In spite of this gap we can use these data together with theothers. For a brief reference, all observations used in this paperare listed in Table 1.

The sampling rate and the compactness of the photographicdata make them suitable for a separate Fourier analysis. Be-cause of their concentration around maxima, the photoelectricdata alone are insufficent for a complete analysis. From the pho-toelectric data we can determine only the main and the modu-lation periods, without being able to make a complete fit withthe necessary higher harmonics. The photographic and the pho-toelectric B data can be analysed together which improves thestability of the fit, and enables us to compute the rate of periodchange. The whole data set spans fifty years and covers all phasevalues of the fundamental period.

3. Frequency analysis

3.1. The photographic data

At this first stage of the analysis we use the standard Fouriertechnique as implemented by Kollath (1990). More than half

A. Nagy: Studies on amplitude modulated RR Lyrae stars. II 441

Table 1.Distribution of data

Year JD-2 400 000 Colour NB Accuracy Source1938−44 28972−31240 Phg. 2412 0.036(1) Oosterhoff (1946)1958 36314−36318 B,V 102 0.01(2) Spinrad (1959)1959 36685−36728 B,V 199 0.01(3) Kanyo (1986)1965 38826−38878 B,V 56 0.02(4) Fitch et al. (1966)1968 39649 B,V 68 0.02(1) Stepien (1972)1971-78 41056−43967 B,V 1387 0.01(3) Kanyo (1986)1987 46948−46949 B,V 49 0.005(4) Jones et al. (1988)

(1) mean error of the magnitude determination(2) mean error of the tie-in of the comparison(3) mean error of the individual observations(4) mean error estimated by the original authors

d

d/2

-2 0 2

.2

.4

.6

.8

1

Am

plitu

de

y

-.01 0 .010

1

Fig. 1. Spectral window of the photographic data. The origin of themain alias features are denoted by the following symbols:y – yearly;d – daily;d/2 – half daily;P0 – periodic. Inset shows the fine structureof the peaks

Table 2. The highest peaks in the amplitude spectrum of the photo-graphic data

n fn[d−1] fn/(n + 1)0 2.65015 2.650151 5.30031 2.650162 7.95045 2.650153 10.6006 2.65015

of the photographic data cover only the ascending branch. Thedescending branch is covered uniformly by the rest. This asym-metry of the data distribution and the daily sampling result ina strong aliasing in the frequency spectrum. Fig. 1 shows thespectral window and Fig. 2 the amplitude spectrum. For thesake of clarity in presenting the spectra we usecompressed plot-ting, i.e., we divide the spectrum in bins of0.005d−1 and wepick the maximum value in each bin. The position of the highestpeaks are listed in Table 2.

1d 1d 1d 1d 1d 1d 1d

0 2 4 6 8 10

.1

.2

.3

.4

.5

Am

plitu

de

Fig. 2.Amplitude spectrum of the photographic data. The daily aliasesare indicated by horizontal arrows

We find the optimum value off0 through a nonlinear leastsquares method by using the first five harmonics off0. Thisyields f0 = 2.6501534 d−1. To search for additional compo-nents we followed the standard method of prewhitening. Aftersubtracting the Fourier sum with thef0, 2f0, . . . 11f0 compo-nents, the residuals were further analysed. The highest peaksare located around the zero frequency component, the basicfrequencyf0 and its harmonics. Fig. 3 shows the prewhitenedspectra around the above mentioned components.

There are four major peaks located at±0.0019d−1 and±0.0046d−1, apparently symmetrically around each compo-nent. The difference between these values is0.0027d−1 =1 y−1, therefore, they are probably aliases. To decide whichcomponent is the alias and to check the degree of symmetry wescan the standard deviation as a function of the modulation fre-quencyfm. The Fourier fits include the following frequenciesf0, 2f0, 3f0, 4f0, fm, f0+fm, 2f0+fm, 3f0+fm, 4f0+fm.Thefm component is changed from−0.005d−1 to 0.005d−1.As can be seen in Fig. 4, the minima of the dispersion are locatedfairly symmetrically, indeed. More precisely, the minima arelocated at−0.001896d−1; 0.001884d−1 and−0.004615d−1;

442 A. Nagy: Studies on amplitude modulated RR Lyrae stars. II

-.01 -.005 0 .005 .010

.10

.10

.10

.10

.10

.10

.10

.1

Fig. 3.Amplitude spectra of the photographic data around the frequen-cies shown in the upper right corners of each panel. The data havebeen prewhitened by thef0, 2f0, . . . 11f0 components. The spectraare vertically aligned for easier comparison

0.004618d−1. Assuming exact symmetry, we fit the frequenciesf0, . . . 4f0, fm, f0 ± fm, . . . 4f0 ± fm with varying fm. Theresult is shown in Fig 5: the standard deviation is the smallestatfm = 0.001886d−1.

The above analysis shows that the photograpic data can berepresented by the following form:

m(t) = A0 +n∑

k=1

[A0(k) sin

(2πkf0(t − t0) + φ0(k)

)+

+A+(k) sin(2π(kf0 + fm)(t − t0) + φ+(k)

)+

+A−(k) sin(2π(kf0 − fm)(t − t0) + φ−(k)

)]+

+A+(0) sin(2πfm(t − t0) + φ+(0)

), (1)

wheref0 = 2.6501534 d−1, fm = 0.001886 d−1, n = 11.The standard deviation of the fit is0.062 mag. Compared withthe value shown in Fig. 5, please note the significant decreaseof the dispersion due to the larger number of harmonics used inthis fit.

-.004 -.002 0 .002 .004

.11

.12

.13

.14

Fig. 4. Standard deviation of the Fourier fits withf0, 2f0, . . . 4f0; fm, f0 + fm, 2f0 + fm, . . . 4f0 + fm com-ponents

0 .02 .04 .06 .08 .1.09

.1

.11

.12

.13

.14

0 .002 .004

.1

.12

.14

Fig. 5. Standard deviation of the Fourier fits withf0, 2f0, . . . 4f0; fm, f0 ± fm, 2f0 ± fm, . . . 4f0 ± fm com-ponents. Inset shows the strongest minima

After subtracting the Fourier sum given by Eq. (1) the resid-uals are further analysed. The resulting amplitude spectrum isshown in Fig. 6. We use compressed plotting with frequencybins of0.005d−1. Although there remained some regularity inthe spectrum, its low level, and the absence of significant peaksshow that there is no remarkable periodicity in the residuals.

3.2. The photoelectric data

The photoelectric data are very extensive and accurate both inV and B, but the sampling is not as good as in the case of thephotographic data. The distribution of peaks in the amplitudespectrum follows a very similar pattern as in the case of thephotographic data. The highest peaks are atk · 2.6501365 d−1,which shows that the basic frequency differs significantly fromthe one obtained through the analysis of the photographic data.After a prewhitening, the search for the modulation frequencyis performed in the same manner as in the case of the photo-

A. Nagy: Studies on amplitude modulated RR Lyrae stars. II 443

0 2 4 6

.01

.015

.02

.025

Am

plitu

de

Fig. 6. Amplitude spectrum of the residuals of the photographic data.The Fourier sum of Eq. (1) is used for the prewhitening

d

d/2

-2 0 2

.2

.4

.6

.8

1

Am

plitu

de

y

-.01 0 .01

1

Fig. 7.Spectral window of the combined photographic and photoelec-tric B data. Symbols are the same as in Fig. 1

graphic data. The result agrees with the one obtained from thephotographic data:fm = 0.001886d−1.

3.3. The combined photographic and photoelectric B data

One can analyse the photographic and the photoelectric B datatogether assuming that there is only a constant shift betweenthem. We can get this constant magnitude difference by itera-tion. First we extrapolate the Fourier sum of Oosterhoff’s pho-tographic data and shift the photoelectric data set in magnitudeuntil we get the best fit to the prediction. In the next step weuse the transformed data in a joint Fourier fit with the photo-electric data. By using this combined data set we get a betterapproximation for the basic frequencyf0. The whole procedureis repeated with this new frequency. After a few iterations thesolution converges. Finally, it turns out that by adding 10.616mag to the photographic data published by Oosterhoff, bringsboth these and the B data in a good agreement.

The combination of the photographic and the photoelectricB data yields a fifty-year-long data set. In spite of the long gapswe can use this set in a Fourier analysis. These data cover all

phase values but the concentration on the ascending branch isstronger than that of the photographic data alone. This problemis visible if we compare the spectral windows shown in Figs. 1and 7. Repeating the same analysis as in Sect. 3.1, we getf0 =2.6501422 d−1 andfm = 0.001886 d−1. The basic frequencyf0 is between the ones obtained during the separate analyses ofthe photographic and photoelectric data, while the modulationfrequencyfm is equal to the preceding values. A least-squaresfit is made with these frequencies in the form of Eq. (1) withn = 9. The standard deviation of the residuals is0.097 mag,which ismuch higherthan the mean error of the observations.This indicates that the combined data set still contains hiddensignificant details of the light variation.

4. Non-stationary frequency analysis

The basic frequencies of the photographic and the photoelec-tric data differ significantly. This fact, together with the highdispersion of the residuals of the combined data set, suggeststhat the basic frequency changes, while the modulation fre-quencyfm (which was the same for both parts of the dataset) remains constant during the whole time span. Based onthe the O – Ccurve of Kanyo (1986) we can assume that theperiod change is linear. Therefore, the basic frequency takesthe form off0 = (P0 + β t)−1, whereβ is the rate of pe-riod change to be determined in the subsequent analysis. Wesearch for the best values ofP0 and β by computing non-stationary Fourier fits with various (P0, β) pairs. We includethef0, . . . 4f0, fm, f0 ± fm, . . . 4f0 ± fm components. Fig. 8displays theσ−1 (P0, β) surface in the close neighborhoodof the optimum solution. For a better visibility of the signifi-cance ofβ, we show thebestσ as a function ofβ in Fig. 9.It is clearly seen that the best non-stationary fit is consider-ably better than the best stationary fit. The best fit is obtainedat the following values:P0 = 0.37733726 ± 3 × 10−8 d andβ = 7.0 × 10−11 ± 0.2 × 10−11 where the errors correspondto the1σ ranges obtained through a Monte Carlo estimation.

The least-squares fit with the time dependent frequenciesis shown in Fig. 10. The amplitude and phase values with theirstandard deviations are listed in Table 3. The errors are computedby assuming that the value ofβ is exact. Changingβ between itserror limits does not influence the errors of the amplitudes andphases. The standard deviation of the residuals is0.064 mag,about two-thirds of that of the constant period case. It can beseen that the mathematical description is fairly accurate, eventhe last two tracks (which were obtained eight years after thepreceding ones) are fitted well. In Fig. 11 we show some ofthe most striking examples for the improvement caused by thenon-stationary fit.

After prewhitening by the Fourier sum given by Eq. (1),the residuals are further analysed. Although there seem to be noprominent peaks present in the prewhitened amplitude spectrum(Fig. 12), several additional (not necessary physically meaning-ful) components might be hidden close to the noise level.

In spite of the good fit and the low amplitude level of theprewhitened spectrum, the standard deviation is rather high (see

444 A. Nagy: Studies on amplitude modulated RR Lyrae stars. II

β

P0

σ-1

0

10-9.37733950

.37732750

Fig. 8. Dependence of the reciprocal of the standard deviation of thenon-stationary Fourier fit onβ at differentP0 values

0 5 10 15

.08

.09

.1

.11

Fig. 9. Dependence of the standard deviation of the best fits onβ.Eq. (1) withf0 = (P0 + βt)−1, fm = 0.001886 d−1, n = 4 is used

Table 1 for the expected errors of the data). We do not know thereason of this excessive scatter. It is unlikely that the wholeamount is due to observational errors. We guess that at least apart of it is a result of some intrinsic noise, similarly to that ofRV UMa (Kovacs, 1995).

5. Comparison of RS Boo and RV UMa

The mathematical description of the light curves of these twostars are essentially the same. Both stars have only one modula-tion frequency component in addition to the basic frequency andthere is no sign ofk f0 ± j fm(j ≥ 2) components. In a deepercomparison we see two differences in the light variations. First,in the case of RS Boo, the basic frequency slowly decreases,while the modulation frequency remains constant during thefifty-year-long time span. The frequencies of RV UMa do notchange detectably from 1956 to 1964.

Table 3. Fourier parameters with their standard deviations for thecombined photographic and photoelectric B data. Eq. (1) is usedwith f0 = (P0 + βt)−1, P0 = 0.37733726 d, β = 7.0 × 10−11,fm = 0.001886 d−1, n = 11 andt0 = 2428872.6330

f A σA φ σφ

0 10.796 0.009f0 0.518 0.019 3.119 0.020

2f0 0.275 0.007 2.240 0.0443f0 0.154 0.015 1.748 0.0854f0 0.080 0.015 1.109 0.0915f0 0.044 0.008 0.290 0.3766f0 0.025 0.007 5.773 0.2397f0 0.011 0.008 5.395 0.9478f0 0.008 0.005 4.456 0.7659f0 0.004 0.004 1.856 0.872

10f0 0.002 0.001 1.545 0.83611f0 0.006 0.003 3.123 0.538

f0 + fm 0.035 0.007 2.343 0.456f0 − fm 0.053 0.008 1.364 0.168

2f0 + fm 0.039 0.009 0.990 0.5232f0 − fm 0.048 0.013 0.989 0.1773f0 + fm 0.032 0.003 0.606 0.3433f0 − fm 0.030 0.012 0.143 0.6284f0 + fm 0.022 0.005 0.197 0.6854f0 − fm 0.023 0.006 5.526 0.4115f0 + fm 0.014 0.004 5.659 0.3405f0 − fm 0.026 0.011 5.591 0.4626f0 + fm 0.015 0.006 5.209 0.1816f0 − fm 0.009 0.004 4.870 0.8977f0 + fm 0.006 0.003 4.487 1.3297f0 − fm 0.012 0.008 4.142 0.5738f0 + fm 0.001 0.004 2.749 2.2108f0 − fm 0.006 0.004 2.978 1.0959f0 + fm 0.002 0.002 2.515 0.4889f0 − fm 0.004 0.002 3.499 0.747

10f0 + fm 0.003 0.001 5.195 0.63410f0 − fm 0.003 0.002 2.081 0.90411f0 + fm 0.003 0.002 0.799 1.15911f0 − fm 0.005 0.001 2.387 0.756

fm 0.011 0.012 2.446 1.092

Second, by examining the shape of the folded light curves(Fig. 13) we see that in the case of RV UMa there is a well-defined node on the ascending branch. This feature is absent forRS Boo. As we show in the next section and in the Appendix, theappearance of the node is tightly connected with the symmetryof the amplitudes of the modulation componentsk f0 ±fm. ForRS Boo (and also for AH Cam — see Smith et al. 1994) thesecomponents are asymmetric and the light curves are withoutnodes.

6. Folded light curves

As we show in the Appendix, analytic solution can be derivedfor the existence of nodes on the folded light curves for thesimple case of a single modulated sinusoidal. In this case the“strength” of the node can be characterized by the following

A. Nagy: Studies on amplitude modulated RR Lyrae stars. II 445

28972

28977

29326

29373

29380

29694

29720

29726

29730

29751

29752

29755

29759

30066

30071

30072

30092

30114

30117

30162

0 1

Phase

30168

30442

30443

30462

30463

30465

30466

30486

30488

30514

30517

30531

30790

30791

30795

30812

30849

31144

31172

31200

31223

0 1

31240

36314

36315

36316

36317

36318

36685

36697

36728

38826

38844

38846

38878

39649

41056

41076

41084

41810

41835

42120

42128

0 1

42443

42449

42454

42465

42493

42532

42786

42870

42899

42905

42910

42927

42947

42964

43207

43228

43251

43276

43288

43298

43302

0 1

43304

43308

43580

43583

43596

43597

43599

43608

43645

43659

43716

43727

43730

43744

43931

43954

43957

43967

46948

0 1

46949

Fig. 10.Fourier fit to the combined photographic and photoelectric B data. Eq. (1) is used with the following parameters:f0 = (P0 + βt)−1,P0 = 0.37733726 d, fm = 0.001886 d−1, n = 11. The vertical tick in the upper left corner denotes one magnitude, numbers showHJD−2400000

446 A. Nagy: Studies on amplitude modulated RR Lyrae stars. II

36316

36685

36728

46948

0 1

46949

Phase0 1

Phase

Fig. 11.Examples for the difference between the stationary (left panel)and the non-stationary (right panel) Fourier fits. The vertical tick de-notes one magnitude, numbers show HJD−2400000

formula

S =|b − 1|b + 1

, (2)

whereb is the ratio of the amplitudes of thef0±fm components,S is thesmoothnessof the folded light curve, which is defined asthe ratio of minimum to maximum thickness. Whenb is equalto 1 the smoothness is zero. This means that there are nodeson the folded curve. Whenb differs significantly from unity,Sincreases to 1, which means that the folded curve is uniformlythick at all phase values. In the Appendix we give a more detailedformulation of this aspect of the light curve topology and theFourier decomposition.

One can perform numerical simulations to find the probabil-ity of the existence of nodes. We generate synthetic light curvesusingkf0, kf0 ± fm (k = 1, . . . , 9). The phase of thef0 com-ponent is set equal to zero. The amplitudes and phases of thefirst five components(Ak, φk) are randomly chosen from theobserved ranges of RRab stars (Jurcsik and Kovacs, 1996). Theother phases(φk+, φk−, φk, k > 5) are random in the [0; 2π]interval. The amplitudesAk (k > 5) follow the pattern of RSBoo with a 25% random fluctuation. The asymmetryA1+/A1−varies from 0.5 to 1.0 The values ofAk+/Ak− for k > 1 areequal toA1+/A1− with a random perturbation of 50%. All ran-domly chosen phases and amplitudes are distributed uniformlyin the prescribed ranges.

On the basis of105 test curves, we study the probabilityof the appearance of nodes as a function of the amplitude ra-tio A1+/A1−. The folded curves are divided into 300 bins andwe examine the thickness (i.e. the difference of the maximumand minimum values) in each bin. The bins where this value is

0 2 4 6.01

.015

.02

.025

Am

plitu

de

Fig. 12. Prewhitened amplitude spectrum of the combined photo-graphic and photoelectric B data. Eq. (1) withn = 11, f0 =(P0 + βt)−1 is used for prewhitening

11.5

11

10.5

10 RV UMa

mag

nitu

de

0 .2 .4 .6 .8 1

11.5

11

10.5

10RS Boo

phase

mag

nitu

de

Fig. 13. Light curves folded with the fundamental periods of twoBlazhko stars

not more than one tenth of the average thickness are accepted asnodes. The result of these simulations (Fig. 14) confirms our ex-pectations from the analytic result obtained for the case of thesimple modulated component. The probability of the appear-ance of a node increases as the amplitudes of the modulationcomponents become more symmetric. The likelihood of the ap-pearance of more than two nodes is less than 0.5%.

A. Nagy: Studies on amplitude modulated RR Lyrae stars. II 447

0

1

2

.5 .6 .7 .8 .9 10

.2

.4

.6

.8

1P

roba

bilit

y

Fig. 14. Probability of the appearance of the nodes as a function ofthe amplitudes of thef0 ± fm components. Number of the nodes areindicated at each curve

7. Conclusion

RS Boo is a very interesting member of the Blazhko-type stars.Its basic period is much shorter than the typical value of theRRab stars. In contrast, its modulation period is the longestamong the known Blazhko-type stars. Therefore, it is interestingto perform a more detailed analysis of the light variation of thisspecial object. The results of this paper can be summarized asfollows:

— The basic period is0.37733726 d, which increases at a rateof β = 7.0×10−11. This value is in the order of magnitudeof the expected evolutionary period changes (Sweigart andRenzini, 1979).

— The 530 d modulation period remains constant during thetotal time span analysed. There is no sign of harmonics ofthe modulation i.e. thekf0 ± jfm, j ≥ 2 components seemto be absent.

— There is no sign of other modulations. We have not foundany period at or near62 d, as mentioned by Kanyo (1986).

— The standard deviation of the fit (0.064 mag) is higher thanthe mean error of the observations. There is an extra scat-ter in the data which cannot be described by our Fourierdescription.

— There is no node on the folded light curve. We pointed outthe existence of a relation between the symmetry of the am-plitudes of thekf0 ± fm components and the shape of thefolded light curves. Weaker symmetry means lower prob-ability of the appearance of nodes. For RS Boo the com-ponents are strongly asymmetric which results in a fairlyuniform folded light curve.

Acknowledgements.I am grateful to Geza Kovacs, Johanna Jurcsik andLaszlo Szabados for the helpful comments and suggestions. I wouldlike to thank the Director and the staff of the Konkoly Observatoryto facilitate my PhD studies. I thank Sandor Kanyo for the clarifyingdiscussions concerning his data. This work has been supported by thefollowig grants:otka t–024022 andakp 9612–412 2,1.

Appendix A: single component modulation

One can derive the analytic condition for the existence of nodeson the folded curves in this very simple case. Here we have onebasic frequencyf0 with additionalf0 ± fm components. Thefollowing notations and simplifications are used: the epocht0and the phaseφ0 of the basic component are set equal to zero;the amplitudes are normalized to that of the basic component;the basic frequency is set equal tof0 = (2π)−1. The ratio of theamplitudes of the modulating waves is denoted byb = A+/A−.We introduce the angular modulation frequencyΩ = 2πfm.

The light curve can be represented by the following expres-sion

m(t) = sin t + A+ sin((1 + Ω)t + φ+

)+

+ A− sin((1 − Ω)t + φ−

)(A1)

The folded curve is obtained by substitutingt = ϕ + k2π (k =1, 2, . . .), whereϕ is the folding phase. Thethicknessd of thefolded curve atϕ can be defined as

d2(ϕ) = limN→∞

1N

N∑k=1

(m(ϕ) − m(ϕ)

)2, (A2)

wherem(ϕ) is the mean (i.e averaged over the modulation pe-riod) value ofm(ϕ), in this case this issin(ϕ+k2π), i.e.,sinϕ.Substitutingm(ϕ) and m(ϕ) into Eq. (A2) and using sometrigonometric relations we get

d2 = limN→∞

1N

N∑k=1

A2+ sin2((1 + Ω)ϕ + Ωk2π + φ+

)+

+ limN→∞

1N

N∑k=1

A2− sin2((1 − Ω)ϕ − Ωk2π + φ−

)+

+ limN→∞

1N

N∑k=1

A+A− cos(2Ωϕ + Ωk4π + φ+ + φ−)−

− limN→∞

1N

N∑k=1

A+A− cos(2ϕ + φ+φ−). (A3)

Calculating the above limits, dividing both sides of the equationby A+A− we get

d2

A+A−=

12

(b +

1b

)− cos(2ϕ + φ+ + φ−). (A4)

We have nodes on the folded curve whend = 0. It follows thatthe condition for the existence of nodes is

12

(b +

1b

)= cos (2ϕ + φ+ + φ−). (A5)

The left-hand side of the equation is always greater or equal to1, the right-hand side is smaller or equal to 1. Equality means

448 A. Nagy: Studies on amplitude modulated RR Lyrae stars. II

thatb = 1, that isA+ = A−. Therefore, we have two nodes atthe phase values

ϕ1 = − φ+ + φ−2

,

ϕ2 = π − φ+ + φ−2

. (A6)

If b /= 1, Eq. (A5) cannot be satisfied, therefore, we have noreal node. In this case we can find the extrema of the thicknessdand we can define thesmoothnessS as the ratio of the minimumand the maximum values ofd. We get extremum when

sin (2ϕ + φ+ + φ−) = 0, (A7)

which means2ϕ + φ+ + φ− = 0 or π. The thickness at thesevalues is

d =

√12

(b +

1b

)± 1 , (A8)

so the smoothness is

S =

√b + 1

b − 2b + 1

b + 2=

|b − 1|b + 1

. (A9)

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