A photometric study of the southern Blazhko star SS For Unambiguous detection of quintuplet components
Katrien Kolenberg, Elisabeth Guggenberger, Thebe Medupe, Patrick Lenz, Lukas Schmitzberger, Robert Shobbrook, Paul Beck, Boitumelo Ngwato, Jan Lub
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 22 October 2018 (MN L A TEX style file v2.2)
A photometric study of the southern Blazhko star SS ForUnambiguous detection of quintuplet components
K. Kolenberg (cid:63) E. Guggenberger , T. Medupe , P. Lenz , L. Schmitzberger ,R.R. Shobbrook , P. Beck , B. Ngwato , and J. Lub , Institute of Astronomy, T¨urkenschanzstrasse 17, A-1180 Vienna, Austria Astronomy Department, University of Cape Town, South Africa Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT, Australia Theoretical Astrophysics Programme, North West University, Mmabatho, South Africa Leiden Observatory, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands
Received 2008 October 8, Accepted 2008 November 18.
ABSTRACT
We present our analysis of photometric data in the Johnson B and V filter of thesouthern Blazhko star SS For. In parallel, we analyzed the V observations obtainedwith the ASAS-3 photometry of the star gathered between 2000 and 2008. In the fre-quency spectra resulting from a Fourier analysis of our data, the triplet structure isdetectable up to high order, both in the B and V data. Moreover, we find evidence forquintuplet components. We confirm from our data that the modulation componentsdecrease less steeply than the harmonics of the main frequency. We derived the vari-ations of the Fourier parameters quantifying the light curve shape over the Blazhkocycle. There is good agreement between the spectroscopic abundance and the metal-licity determined from the Fourier parameters of the average light curve. SS For ispeculiar as a Blazhko star because of its strong variations around minimum light. Key words: stars: oscillations – stars: variable: RR Lyrae – stars: individual: SS For– techniques: photometric.
A large fraction of the RR Lyrae stars shows a periodicamplitude and/or phase modulation with a period of typ-ically ten to hundreds of times the pulsation period. Thisphenomenon is referred to as the Blazhko effect, after theRussian astronomer who first reported it (Blazhko 1907).The origin of the Blazhko effect is still a matter of con-troversy. The most widely discussed models attribute theeffect to either the consequences of the interaction of amagnetic field with the main radial pulsation (Shibahashi& Takata 1995), or a resonance between the main modeand a non-radial mode of low degree (Van Hoolst, Dziem-bowski & Kawaler 1998; Nowakowski & Dziembowski 2001;Dziembowski & Mizerski 2004). However, also models thatdo not require nonradial modes have been proposed, e.g. byStothers (2006). This scenario attributes the Blazhko varia-tion to a variable turbulent convection due to transient mag-netic fields in the star. All models presently proposed for theBlazhko effect have shortcomings in explaining the varietyof features shown by Blazhko stars. Therefore, theoretical (cid:63)
E-mail: [email protected] efforts to revise or expand the models would be worthwhile,and even the exploration of alternative explanations.Since its discovery almost a century ago, most studiesof the Blazhko effect were carried out from the northernhemisphere (e.g., Szeidl 1988, and references therein; Smith1995, and references therein). For a long time there werefewer well-established field
Blazhko stars at southern decli-nations. The available data for southern field Blazhko stars(e.g., Hoffmeister 1956; Kinman 1961; Clube et al. 1969;Lub 1977) are often insufficient to determine the Blazhkoperiods with the required accuracy. Accurate and completephotometric data sets of southern field Blazhko stars arestill lacking. Extensive multi-target surveys such as MA-CHO (Alcock et al. 2000, 2003) and OGLE (Moskalik &Poretti 2003), and ASAS (All Sky Automated Survey, Po-jmanski 2000) have significantly contributed to our knowl-edge of the Blazhko effect, and expanded the list of knownBlazhko stars. Wils & Sodor (2005) published a list of newand confirmed Blazhko targets, and Szczygiel & Fabrycky(2007) obtained interesting new results, among which is astar with multiple Blazhko periods.Nevertheless, long-term campaigns dedicated to partic-ular Blazhko stars and yielding complete light curves at dif- c (cid:13) a r X i v : . [ a s t r o - ph ] D ec K. Kolenberg, et al. ferent phases in the Blazhko cycle remain of great scien-tific value. Only these can reveal changes in the light curveshape, as well as long-term changes in the characteristics ofthe Blazhko effect such as amplitudes, phases, and periods.In this way they can give crucial information for decidingamong the different hypotheses for the Blazhko effect.Photometry with a good spread over both the pulsationand Blazhko cycles allows us to perform the first rigorousfrequency analysis of a southern field Blazhko star. The firstaccurate photometric data set covering both the pulsationand the Blazhko cycle was published by Jurcsik et al. (2005a)for the northern short-period Blazhko star RR Gem.SS For (SAO 167572; α ( J h m s , δ ( J ◦
51’ 54”) is a relatively bright southern Blazhko star.According to the General Catalogue of Variable Stars(Kukarkin 2003) its V brightness changes within therange 9.45-10.60. The General Catalogue of Variable Stars(GCVS) lists a period of P = 0.495432 d (Kholopov et al.1998) or about 11 h 53 min, corresponding to a frequencyof f = 2 . − Our photometric observations were carried out with 3 differ-ent telescopes in the southern hemisphere. All were equippedwith photomultipliers to obtain comparative photometry inthe Johnson B and V passbands.The photometric observations of SS For started fromthe South African Astronomical Observatory (SAAO) inSutherland, South Africa, in October 2004, and were car-ried out from the 0.5-m and 0.75-m telescopes until Novem-ber 2005. The 0.5-m telescope was equipped with the mod-ular photometer. At the 0.75-m telescope we used the UCTphotometer. From July until September 2005, the 0.6-mtelescope at Siding Spring Observatory (SSO) near Coon-abarabran, Australia, joined in the observing campaign. Itsphotometer is equipped with a GaAs photomultiplier tube.Given the nearly 12-h period of SS For, the complementarylongitudes of the two observatories allowed for observationsof a larger part of the light curve. Additionally, the heightof 1-day aliases introduced by the spectral window (see alsoFig. 1) was decreased by the use of more than one observingsite. This enables an easier and more accurate determinationof the frequencies of the stellar light variations. The totaltime span of the data set is 345 days, almost 10 Blazhkocycles. Figure 1.
Window Function for the data set gathered at SAAOand SSO (2004-2005).
The observations were carried out using the 3-startechnique, as described by Breger (1993). The comparisonand check stars were HD13334 ( α ( J h m s , δ ( J ◦
06’ 13”, V =9.7, B − V = 0.57) and HD13181( α ( J h m s , δ ( J ◦
01’ 29”, V =9.8, B − V = 0.51). Table 1 shows a journal of the measurements.Depending on the weather conditions, the accuracy of theindividual differential measurements varied between 3 and10 mmag. In the reduction procedure, variable atmosphericextinction was taken into account. The night-to-night shiftsof the differential magnitudes of the two comparison starslaid within the scatter of a few millimags.In order to combine the data sets from the differenttelescopes and instruments we carried out transformationsto the standard system. This is crucial, since for variableswith changing light curve shapes zero point offsets can of-ten not be correctly determined. Moreover, the large varia-tions in the B-V colour (more than 0.3 mag - see Figure 3)throughout the pulsation cycle result in a considerable dis-tortion of the light curves (as large as 0.02-0.03 mag) if thecolour systems are not the same. For the SAAO data weused transformation coefficients determined by D. Kilkenny(private communication), and from our own standard starmeasurements. For the SSO data we took coefficients deter-mined from standard star measurements by R. Shobbrook(private communication), close to the ones given by Bernd-nikov & Turner (2001).The light curves of the new data in Johnson B andJohnson V , folded with the main pulsation period, are shownin Figure 2. The differential magnitudes are shown with re-spect to HD13181. SS For (ASAS J020752-2651.9) was observed as part of theAll Sky Automated Survey (ASAS-3; Pojmanski 2002) fromNovember 2000 onwards. In most cases ASAS photometryis accurate to about 0.03 mag (Pojmanski 2000). We anal-ysed SS For data from ASAS-3 which were gathered be-tween November 2000 and July 2008 (HJD 2451868.613–2454656.892, 484 points over 2788 days or almost 80 Blazhko c (cid:13) , 000–000 photometric study of the southern Blazhko star SS For Table 1.
Observing log for the SS For Johnson B and V data. SAAO data (2004+2005)
SSO data (2005)Night Length [hrs] N Observer Night Length [hrs] N Observer2453288 2.73 18 EG 2453569 0.98 6 PL2453289 7.30 43 EG 2453571 2.36 11 PL2453290 8.57 62 EG 2453574 2.39 10 PL2453292 3.54 24 EG 2453576 2.30 11 PL2453296 6.07 44 EG 2453580 2.61 12 PL2453300 5.92 21 EG 2453581 2.73 13 PL2453301 8.29 60 EG 2453582 2.59 14 PL2453302 3.67 23 EG 2453583 2.66 14 PL2453303 7.07 44 EG 2453590 2.66 12 LS2453304 7.59 51 EG 2453591 2.91 12 LS2453319 4.01 18 TM 2453610 5.76 27 LS2453320 3.74 17 TM 2453611 4.81 22 LS2453321 4.55 19 TM 2453612 1.78 9 LS2453324 3.75 18 TM 2453621 1.81 10 LS2453325 4.13 22 TM 2453622 1.58 9 LS2453367 1.69 6 EG 2453633 3.57 26 BS2453574 4.29 26 EG2453575 4.12 23 EG2453576 2.82 20 EG2453578 0.54 4 EG2453579 3.17 21 EG2453580 4.86 29 EG2453582 4.81 31 EG2453584 4.51 26 EG2453587 2.11 12 PB2453588 1.48 7 PB2453592 3.86 26 EG2453593 5.18 34 EG2453594 5.38 36 EG2453596 5.40 36 EG2453598 5.66 40 EG2453599 5.60 45 EG2453601 5.90 48 EG2453602 6.00 46 EG
SAAO total 2004 2005
SSO total (=2005)Observing time [hrs] 158.32 82.63 75.70 Observing time [hrs] 43.51Number of datapoints 1000 490 510 Number of datapoints 218Number of nights 34 16 18 Number of nights 16Total time span [days] 314 79 28 Total time span [days] 64Total observing time [hrs] 201.83Total number of datapoints 1218Number of nights 50Total time span [days] 345 cycles). The folded V ASAS-3 light curves are shown inFig. 4.
The frequency analyses were performed with Period04 (Lenz& Breger 2005), a package applying single-frequency powerspectra and simultaneous multi-frequency sine-wave fitting.Fourier analyses were carried out on subsets of the photo-metric data in both filters. Before merging them, we anal- ysed each of the standardized data sets (SAAO, SSO, ASAS)separately to check the compatibility of the frequency solu-tions.
The best fits to the data were determined by means ofa successive prewhitening strategy. From this analysis thetriplet structures typical for RR Lyrae Blazhko stars clearlyemerged.For Blazhko stars very often the a priori choice is made c (cid:13) , 000–000 K. Kolenberg, et al.
Figure 2. B (upper panel) and V (lower panel) light curve ofSS For, folded with the main period. The full line represents themean light curve. The Blazhko effect is very pronounced aroundboth minimum and maximum light. Figure 3.
Mean B − V color curve of SS For, resulting fromsubtracting the two mean light curves (Figure 3). Figure 4.
SS For data from ASAS-3, folded with the main pul-sation period.
Figure 5.
Fourier spectrum for the data set gathered at SAAOand SSO (2004-2005). The highest peak corresponds to the mainfrequency at 2.018433 cd − . The 1-day aliasing is clearly visible. to describe the light curve variations by means of equidistanttriplets at the main frequency and its harmonics accordingto the formula (see also Kov´acs 1995): f ( t ) = A + (cid:80) nk =1 [ A k sin(2 πkf t + φ k )+ A k + sin(2 π ( kf + f B ) t + φ k + )+ A k − sin(2 π ( kf − f B ) t + φ k − )]+ B sin(2 π ( f B t + φ B )) . (1)The amplitudes and phases are calculated up to order n , forwhich the amplitudes of the higher-order harmonics are stillabove the significance level. In this formula, f is the mainpulsation frequency and f B is the Blazhko frequency.In order not to presume equidistantly spaced tripletswe also fitted the data with triplet structures without thecondition of equidistance, i.e. the side peak frequencies are”let free”. The resulting departure from equidistance (seealso Breger & Kolenberg 2006) was smaller than the uncer-tainties on the frequencies, so we can resort to Equation (1)for the triplet fits. c (cid:13) , 000–000 photometric study of the southern Blazhko star SS For We calculated optimum values for the frequencies, theiramplitudes and phases by minimizing the residuals of thefit given by Equation (1). Errors on these values were deter-mined through extensive Monte Carlo simulations, and areof the same order as the error values calculated followingBreger et al. (1999) and Montgomery & O’Donogue (1999).Handler et al. (2000) found that correlated noise may leadto an underestimation of the errors. Following their con-clusions we multiplied the errors resulting from our MonteCarlo simulations by a factor of 2, in order to get a reliableand realistic error margin. The uncertainties obtained andgiven in Table 2 were confirmed by our analyses of differentsubsets of the data.
With the ASAS-3 data set until July 2008 we find f =2.01843 ± − for the main frequency and f + f B =2.04712 ± − for the first detected side peakfrequency, resulting in a Blazhko frequency f B =0.02868 ± − . Wils & S´odor (2005) published the results of aperiod analysis of a set of southern RR Lyrae stars exhibit-ing the Blazhko effect and published new elements for pre-viously known as well as new and suspected Blazhko RRabstars of the ASAS database. They determined the Blazhkoperiods of the variables with a fit containing the first fourtrial modulation frequencies, i.e., a fit including the tripletcomponents around the main frequency f and its harmonic2 f . On the basis of the earlier ASAS-3 photometry for SSFor, they list a Blazhko period of 34.8 d, which correspondswell with the value we find.We checked the ASAS data for period changes by look-ing at all normal maxima in the ASAS data. The V data The highest recorded peak-to-peak amplitude in our V data is 1.34 mag. The harmonics of the main radial fre-quency are significant up to 16 f and both triplet compo-nents are detected above the 3 . σ noise level for combinationfrequencies up to the 11th order.When including the ASAS-3 V data in the analysis, weobtain f = 2 . ± . − for the radial pulsationfrequency and f N = 2 . ± . − for the rightside peak frequency. Given the longer time span of the dataset, we use the frequencies found from the combined ASAS-3+SAAO+SSO data as start values for optimization to fitthe new data gathered at SAAO and SSO. Note that we donot use the ASAS data to determine the final fit given inTable 2. The B data RR Lyrae stars have higher amplitudes in the blue re-gion of the spectrum. The highest recorded peak-to-peakamplitude from our B data set is 1.65 mag. For the mainfrequency and its harmonics, the amplitudes in B are on average a factor of about 1.23 larger than those in V for f and its harmonics up to order 10.Since they were based upon a larger and more extendedset of data, we used the optimized frequencies of the fit tothe V data to fit our data obtained in the B filter. Giventhe higher amplitudes in the B filter, the harmonics of f are significant up to a higher order, 17 f . Also both tripletcomponents have amplitudes above the 3 . σ noise level (theadopted threshold for combination frequencies) up to the10th order. The final fit
Table 2 list the results of a multifrequency fit to thecombined set of SAAO+SSO data, according to Equation(1) and based on the magnitude differences relative toHD13181, the comparison star with coordinates closer to SSFor and a smaller colour difference to the target. The resultsbased on the magnitude differences relative to HD13334 dif-fer from the ones published in Table 2 within the given un-certainties. Note that for independent frequencies (in thiscase only f and f B ) to be significant we required a signal-to-noise level exceeding 4. For combination frequencies werequired the signal-to-noise level to be higher than 3.5. Ephemerides
From the new SAAO and SSO data we derived the fol-lowing ephemerides of maximum light and maximum pulsa-tion amplitude:HJD( T max ) = 2453296 .
440 + 0 . × E pulsation (2)HJD( T Blmax ) = 2453298 .
914 + 34 . ± . × E Blazhko (3)The time of maximum light ( T max ) was recorded in ourdata. The time of the highest light curve amplitude ( T Blmax )was determined on the basis of our best fit to the data.Our highest recorded maximum was at HJD=2453296.440but according to our best fit it preceded the absolutemaximum, which happened 5 pulsation cycles later atHJD=2453298.914.
After subtraction of the best triplet fit to the data, there isstill residual scatter of the light curve (Figure 6, grey filledcircles), concentrated at the phases of minumum to maxi-mum light (pulsation phase φ = 0 . − .
1) and in the bumpphase (pulsation phase φ = 0 . − . B data in which themodulation components have the higher amplitudes, we firstdetected a peak at the frequency 6.1125 cd − with a signifi-cant amplitude. This peak is located at the expected positionof the higher quintuplet component at 3 f , 3 f + 2 f B . Wealso detected significant power at the following frequencies:12.1678 cd − (6 f + 2 f B ) and 10.0349 cd − (5 f − f B ). c (cid:13) , 000–000 K. Kolenberg, et al.
Table 2.
Amplitudes and phases (fraction of 2 π ) of the pulsation and modulation frequency components of SS For (triplets) for the bestfit including triplet components. The values displayed in italics correspond to combination frequencies not exceeding a signal-to-noiselevel of 3.5. The residuals of the triplet fit to the V data are 0.017 mag, to the B data 0.019 mag. At the bottom of the table we list themost prominent quintuplet components that are found in the data set. After inclusion of the 3 prominent quintuplet components theresiduals are 0.015 mag both in B and V . The uncertainty on f is 10 − cd − and the uncertainty on f + f B is 4 × − cd − . f [cd − ] A V [mag] φ V [ rad2 π ] σ ( φ V ) A B [mag] φ B [ rad2 π ] σ ( φ B ) ± . ± . f f f f f f f f f f f f f f f f f f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f + f B f − f B f B f + 2 f B f + 2 f B f − f B (cid:13) , 000–000 photometric study of the southern Blazhko star SS For Figure 6.
Residuals of the B (left panel) and V (right panel) data of SS For after subtraction of the mean light curve (top panels, opencircles) and after subtraction of the best fit including triplets (top and bottom panels, filled grey circles). When including the quintupletcomponents in the fit (bottom panels with a larger magnitude scale, filled black circles), the residuals reduce even more. Figure 7.
Fourier spectrum showing the quintuplet componentsfound in the data for SS For after subtraction of the main fre-quency. its harmonics and the triplet components. The notation” q k + ” and ” q k − ” stands for kf +2 f B and kf − f B respectively;”a” denotes alias peaks. separations within the uncertainty on the frequency. Asit is quite remarkable to find additional frequencies exactlyat the expected positions of quintuplet components, we haveadded the found frequencies, their amplitudes and phases Figure 8.
Fourier spectrum for an additional frequency compo-nent found in the B data after subtraction of the triplet fit plusthe found quintuplet components. Left panel: Spectrum between0-20 cd − , right panel: zoom. The peak is at the expected positionof a septuplet component (6 f + 3 f B ). below Table 2. Figure 7 shows the Fourier spectrum in the B data after subtraction of the triplet solution. Quintupletcomponents and aliases are indicated. We clearly see evenmore quintuplet components in the Fourier spectrum thanwere obtained through subsequent prewhitening. The quin-tuplet components emerge most clearly from the B data set,but we find the same frequencies with significant amplitudesin the V data.After including the detected quintuplet components inthe fit, we find residual power at several frequencies aroundthe significance level. None of them is clearly a quintupletcomponent and at this level spurious peaks occur. More c (cid:13) , 000–000 K. Kolenberg, et al. quintuplet components may be present in the data, but werenot unambiguously detected by us. Given the imperfect cov-erage of our data (gaps) we considered it inappropriate to fitmore quintuplet components to the data. We note, however,that in the B data we find evidence for a significant peakat 12.1964 cd − , corresponding to (6 f + 3 f B , see Figure 8).More evidence is needed to confirm this frequency as one re-ally present in the data and not an artefact of the sampling.Jurcsik et al. (2008) identified frequencies at kf ± f B indata of MW Lyr. They also found septuplet frequencies (i.e.,frequencies at kf ± f B ) and kf ± f B components in theresiduals of their data set, though much less significantly.Multiplet components kf ± jf B with j (cid:62) The properties of the modulation components kf + f B and kf − f B in the frequency spectra of a Blazhko star consti-tute an important test for the models proposed to explainthe modulation. However, it is important to obtain reliableuncertainties for the amplitudes and phases of the modula-tion components. As was shown by Jurcsik et al. (2005a),the amplitudes of the triplet (or generally: multiplet) com-ponents depend strongly on the coverage. From their extensive study comprising 731 Blazhko variablesin the LMC, Alcock et al. (2003) found that the relativeamplitudes of the first order modulation components areusually in the range 0 . < A ± /A < .
3, but there aresome extreme values. For our V data, we find A + /A =0 . ± .
007 and A − /A = 0 . ± . B data, A + /A = 0 . ± .
005 and A − /A = 0 . ± . R k = A kf + f B /A kf − f B (4)and∆ φ k = φ kf + f B − φ kf − f B . (5) We also list the so-called asymmetry parameter Q = A + − A − A + + A − , (6)introduced by Alcock et al. (2003) to quantify the degreeof asymmetry in the peaks. The distribution of the Q pa-rameter for the Blazhko stars from the MACHO data base(Alcock et al. 2003) peaks at +0.3.According to Kov´acs (1995, and references therein), theparameter combinations R k and ∆ φ k are expected to beconstant under the assumption of a simple oblique rotator-pulsator model. However, such a simple model seems inad-equate to explain the variety of observations related to theBlazhko effect, and hence we give Table 3 rather as lightcurve diagnostics (see also Smith et al. 1999 and Jurcsik etal. 2005a).Due to the rather large error bars on the parametercombinations, the phase differences ∆ φ k , especially for the V data, do not deviate significantly from a constant value.On the other hand, the amplitude ratios R k show largechanges, both in V and in B , and with the obtained errorbars they can definitely not be considered as constant.As can be seen in Table 3 the Q value can change itssign in different orders. The positive Q values at multipletorder k = 1, 4 and 5 point to an asymmetry with higheramplitudes at the higher frequency lobes in the triplets, asis mostly the case in Blazhko stars. At multiplet orders k =2 and 3 the left side peaks are higher, yielding negative Q values. B versus V The Blazhko modulation in the B filter is larger than in the V filter. Jurcsik et al. (2005b) found, from a combinationof different data sets, that the value of A mod ( B ) /A mod ( V )lies within the range of 1.23 – 1.39 with a mean valueof 1.30. Their value was determined from the sum of theFourier amplitudes of the first four modulation components, A f + f B , A f − f B , A f + f B , A f − f B . For our data we obtaina value A mod ( B ) /A mod ( V ) = 1 . ± .
10, taking into ac-count the first four modulation components. At 2 f + f B the side peak amplitude determined from our data set israther low. When the higher order side peaks are taken intoaccount the value of A mod ( B ) /A mod ( V ) does not increasesignificantly (see Table 4). This implies that the modulationis indeed stronger in B than in V , and by the same factoras holds for the main frequency and its harmonics (Section3.2.1). Jurcsik et al. (2005a) reported that the decrease in am-plitude of the successive harmonics of f is much steeperthan that over the side peaks. Whereas the latter decreasecan be described as exponential, the side peak amplitudeshave a more linear decrease. As a consequence, the relativecontribution of the modulation at higher orders is larger.This is illustrated in Figures 9 and 10, showing the am-plitude ratios A ( f k ) /A ( f ), A ( f k + f B ) /A ( f + f B ), and A ( f k − f B ) /A ( f − f B ) for the V and the B data respec-tively. The amplitudes of the left and right side peaks de- c (cid:13) , 000–000 photometric study of the southern Blazhko star SS For Table 3.
Side lobe amplitude ratios R k , phase differences ∆ φ k (fraction of 2 π ) and asymmetry parameters Q as defined in the text, andtheir respective errors for the V and B data. k denotes the multiplet order. k R k ( V ) σ R k ( V ) ∆ φ k ( V ) σ ∆ φ k ( V ) Q ( V ) σ Q ( V ) R k ( B ) σ R k ( B ) ∆ φ k ( B ) σ ∆ φ k ( B ) Q ( B ) σ Q ( B )1 2.05 0.28 0.15 0.02 0.34 0.07 2.09 0.24 0.14 0.02 0.35 0.062 0.43 0.24 0.18 0.07 -0.40 0.31 0.26 0.16 0.25 0.09 -0.58 0.503 0.53 0.19 0.19 0.04 -0.30 0.16 0.54 0.17 0.17 0.04 -0.30 0.144 1.69 0.51 0.12 0.04 0.26 0.11 1.68 0.39 0.12 0.03 0.25 0.085 1.18 0.36 0.11 0.04 0.08 0.04 1.08 0.27 0.09 0.03 0.04 0.016 1.13 0.46 0.13 0.05 0.06 0.03 1.00 0.32 0.12 0.04 0.00 0.00 Table 4.
The ratio of the strength of the modulation componentsin the Johnson B and V filter for increasing order k . k = 1 takesinto account the side peak amplitudes around f ,..., kf .Order k A mod ( B ) A mod ( V ) A mod ( B ) /A mod ( V )1 0.091 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Figure 9.
Amplitude ratios A ( f k ) /A ( f ), A ( f k + f B ) /A ( f + f B ),and A ( f k − f B ) /A ( f − f B ) of the detected frequencies for the V data. The amplitudes of the main frequency and its harmonics de-crease exponentially with increasing order. The amplitudes of theleft and right side peaks, on the other hand, decrease less steeplyand show more irregular behavior at lower multiplet orders. crease less steeply. They also show more irregular behaviorat lower multiplet orders ( k = 2 , , Figure 6 shows the residual light curve of SS For folded withthe main pulsation period after subtraction of the mean light
Figure 10.
Amplitude ratios A ( f k ) /A ( f ), A ( f k + f B ) /A ( f + f B ), and A ( f k − f B ) /A ( f − f B ) of the detected frequencies forthe B data. See also caption of Figure 9. curve (open circles). It shows that the Blazhko modulationin SS For is not only concentrated in the minimum to max-imum phase interval, as is the case for other Blazhko stars(see, e.g., Jurcsik 2005a). The strong variations in the bumpphase before minimum light ( φ = 0 . − .
8, see also Guggen-berger & Kolenberg 2006) are reflected in the scatter. As isknown for Blazhko variables, the scatter is generally largestaround the phase of maximum light. For SS For, there is acomparable scatter around minimum light. A large modula-tion around minimum light was also detected in the Blazhkostar MW Lyr (Figure 3 in Jurcsik et al. 2008).
In order to assess the light curve variations over the Blazhkocycle, our data were divided into 10 subsets. Each subsetcorresponds to a 0.1 phase interval over the Blazhko cycle.For the ephemeris of Blazhko phase Ψ = 0, defined as thephase of the brightest maximum, we adopted the time of thehighest detected maximum obtained from the best fit to ourobservations, i.e., T = 2453298 . B data the gapsin the light curves at different Blazhko phases do not allowus to perform a fit for each of the phase intervals (Figure 14,available online). If coverage of minimum or maximum lightis lacking, it is impossible to reliably fit the data and de- c (cid:13) , 000–000 K. Kolenberg, et al.
Table 5.
A log of the subsets of the observations used to construct the light curves at different phase intervals in the Blazhko cycle. Forthe V data set we used SAAO, SSO and ASAS data, for the B filter only SAAO and SSO.∆Ψ HJD (-2450000) ( V ) N ( V ) HJD (-2450000) ( B ) N ( B )0.0-0.1 1868.61332 – 4629.92251 201 3296.440–3612.133 1320.1-0.2 1870.60972 - 4527.50794 248 3300.380–3581.324 1840.2-0.3 1873.60768 - 4633.91429 85 3303.413–3584.678 1740.3-0.4 1878.59277 - 4534.49935 84 3587.590–3622.167 380.4-0.5 1880.57795 - 4501.57064 133 3590.204–3591.321 240.5-0.6 1884.56548 - 4645.90353 163 3592.519–3594.678 960.6-0.7 1888.56074 - 4649.89091 241 3319.402–3633.294 1370.7-0.8 2067.93681 - 4652.92994 209 3288.278–3602.671 2040.8-0.9 1930.5492 - 4656.89163 110 3289.481–3571.323 1580.9-1.0 1900.53501 - 4520.53379 206 3296.257–3574.686 66 termine Fourier parameters. For the V data, however, weadded the ASAS-3 photometry, and fitted each of the phaseintervals with a higher-order harmonic fit according to: f ( t ) = A + (cid:88) k =1 A k sin(2 πkf k t + φ k ) , (7)and calculated Fourier parameters, namely the amplituderatios R k = A k /A (8)and the epoch-independent phase differences φ k = φ k − kφ . (9)These parameters offer a way to quantify the shape of thelightcurves. Fig. 12 shows the variations of the amplitudes A k and the phases φ k for k = 1 , ...,
4, and Figure 13 the de-rived Fourier parameters. The Fourier parameters are alsolisted in Table 6. It is clear that the first order amplitude A decreases towards minimum Blazhko amplitude more dras-tically than do the higher order amplitudes A , A , A . Asa consequence, the R k parameters reach their maximumamplitude around minimum light. Small variations in the φ k , and subsequently the φ k φ is the largest, as was alsonoted by Jurcsik et al. (2008).Our observed Fourier parameters for SS For clearly fallwithin the ranges of Fourier parameters derived from the V light curves of 257 RRab stars published by Kov´acs &Kanbur (1998). The ranges we obtain also intersect with thetheoretical parameters calculated by Feuchtinger (1999) andDorfi & Feuchtinger (1999, their Figure 8 and 11). Note thatthe latter authors express their Fourier parameters in thecosinus frame. For the amplitude ratios we have values in therange R = 0 . − . R = 0 . − . R = 0 . − . V data. The phase differences in the cosinus frameare in the ranges φ = 3 . − . φ = 1 . − . φ = − . − . The Blazhko character of SS For was recognized by Lub(1977), who noticed strong variations around minimum lightof the star. At that time, its Blazhko period was not known.
Figure 12.
Variations of the amplitudes A k and the phases φ k over the Blazhko cycle for k = 1 , ..., Wils & Sodor (2005) published a study on Blazhko vari-ability from the ASAS database. SS For appears in theirtable with elements of new and suspected Blazhko RRabstars. They determined a Blazhko period of 34.8 d. Fromour new data set we find a value for the Blazhko period P B = 34 . ± .
05 d, as the difference between f + f B and f . c (cid:13) , 000–000 photometric study of the southern Blazhko star SS For Figure 11. V light data (crosses) in different 0.1 phase intervals of the Blazhko cycle (34.94 ± The observed asymmetry of the modulation components inthe triplets , also observed in SS For (see Section 3.3.1), re-mains unexplained by both the resonance model and themagnetic model, at least in the degree it is observed.In our data we find clear evidence for quintuplet com-ponents in the vicinity of the harmonics of the main fre-quency in our data. A quintuplet structure is generally pre-dicted by the magnetic model (Shibahashi & Takata 1995),where the magnetic field causes the main radial mode to de-form and have additional quadrupole ( (cid:96) = 2) components.For more than a decade after the magnetic model was first presented, no evidence for quintuplet components was foundfrom any data set. The argument given by proponents of themagnetic model was that a quintuplet structure may man-ifest itself as only a triplet depending on the geometricalconfiguration (angles of pulsation axis, magnetic axis andaspect angle). Also, the quintuplet components might havesuch a low amplitude that they are hidden in the noise ofthe frequency spectrum. This is supported by recent findingsby Hurta et al. (2008) in RV UMa, Jurcsik et al. (2008) inMW Lyr, and this work, but it does not imply the magneticmodel is the one to be preferred over the resonance model.A magnetic field of about 1 kG is needed in the mag- c (cid:13) , 000–000 K. Kolenberg, et al.
Table 6.
Variations of the Fourier parameters A , R k , φ , and φ k over the Blazhko cycle for k = 1 , ..., A R R R φ φ φ φ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Figure 13.
Variations of the Fourier parameters R k and φ k over the Blazhko cycle for k = 1 , ..., netic model for the amplitude modulation to be observ-able (Shibahashi & Takata 1995). The only Blazhko star forwhich a dedicated spectropolarimetric campaign has beencarried out so far is RR Lyr, the prototype of the class, andthe results are rather contradictory. Babcock (1958) and Ro-manov et al. (1994) reported a variable magnetic field in RRLyr with a strength up to 1.5 kG, whereas Preston (1967)and Chadid et al. (2004) contradict these measurements.The detection of magnetic fields in RR Lyrae stars has been hampered by the fact that these stars are quite faint; RRLyr is by far the brightest with V = 7.2-8.2. With the newgeneration of powerful spectrographs attached to big tele-scopes, however, many more RR Lyrae stars have now comewithin reach for spectropolarimetry.Interestingly, in this data set we only find clear evi-dence for quintuplet components around the harmonics ofthe main frequency and not around the main frequency it-self. This may be due to the fact that the observed am-plitude spectrum is also influenced by the coverage of ourmeasurements (see Jurcsik et al. 2005a). Satellite data withquasi-uninterruped coverage over several Blazhko cycles willgive an answer to many questions concerning the frequencyspectrum of Blazhko stars, the modulation components andthe occurrence of quintuplet components. Jurcsik & Kov´acs (1996), herafter JK96, determined a P − φ − [Fe/H] relation for fundamental mode RR Lyraestars. Their calibration was based on a sample of 81 RRabfield Blazhko stars, and its application to RR Lyrae starswith independent spectroscopic metallicities has proven thatit has an overall prediction accuracy of 0.12 dex (Kov´acs2005). The question, however, is whether this empirical for-mula is also applicable to amplitude- and phase-modulatedfundamental mode RR Lyrae stars.According to Layden (1994), SS For has a spectroscopiciron abundance [Fe/H] spec = -1.35, which is -1.09 on the scaleof the JK96 relation. Based on ASAS data, Kov´acs (2005)calculated the iron abundance [Fe/H] Four according to theJK96 formula, and found a ∆[Fe/H]=[Fe/H] spec -[Fe/H]
Four of 0.44. Kov´acs (2005) attributed the large deviation bothto an inaccurate spectroscopic iron abundance determina-tion, as well as the light curve shape and therefore Fourierparameter variations due to the Blazhko effect. We supportthe latter explanation, since from our determination of theFourier parameters at different phases in the Blazhko cycle,we obtain values for φ between 4.58 and 5.13 rad (in thesinus frame, as required for the JK96 formula). This leadsto metallicities [Fe/H] Four between -1.55 and -0.81, a ratherlarge range but with an average (-1.18) rather close to thespectroscopic value.Based on the result of Alcock et al. (2003), it is pos-sible that the average light curves of Blazhko stars can beemployed equally well in the empirical formula calibrated c (cid:13) , 000–000 photometric study of the southern Blazhko star SS For on strictly periodic stars. However, long-term monitoring -from months to years - is needed to achieve sufficient cov-erage for Blazhko variables. The test can be done for SSFor, for which we obtained average light curve parameterswith a high accuracy. For the mean light curve, this yieldsthe value φ = 4 . ± .
057 (this corresponds to 0.785 ± π , as can be derived from Table 2). Ap-plication of the JK96 calibration then yields [Fe/H]=-1.07 ± While in most Blazhko stars the variations of the pulsa-tion maximum are most pronounced, in SS For also theminimum shows strong variations during the Blazhko cycle.These variations have already been reported by Lub (1977)and have been investigated in more detail by Guggenberger& Kolenberg (2006) on the basis of the data presented inthis paper. They found a periodic behaviour of the so-calledbump in the light curve that appears just before minimumlight. The bump occurs at an earlier phase around Blazhkominimum and later around Blazhko maximum. Also thestrength of the bump shows a dependence on the Blazhkophase: the bump is most distinct during minimum and al-most vanishes during Blazhko maximum.The bump has been explained by a collision betweenthe free-falling high atmospheric layers and the deep atmo-sphere, which has a smaller infall velocity (Gillet & Crowe1988). From spectroscopic data of RR Lyr, Preston, Smak, &Paczynski (1965) concluded that a displacement of the shockforming region occurs over the Blazhko cycle. This may in-duce an earlier or later occurence of the bump in the lightcurves. Also the U-B excess observed on the rising branch,as well as the strength of the H emission (Hardie 1955), bothlinked to the shocks in RR Lyrae stars, are variable over theBlazhko cycle (Preston, Smak, & Paczynski1965).
High-precision photometric observations in the Johnson B and V filter were obtained from the South African Astro-nomical Observatory (SAAO) and Siding Spring Observa-tory (SSO), Australia, between October 2004 and September2005. The time span of the data covers almost 10 completeBlazhko cycles.A detailed analysis of the combined standardized pho-tometric measurements led to the following conclusions: • From a Fourier analysis of the combined SAAO+SSOdata set we obtain f = 2 . ± . − for theradial pulsation frequency and f N = 2 . ± . − for the right-side peak frequency. • From our data we find a value for the Blazhko period P B = 34 . ± .
05 d. • The amplitudes in the B data are a factor of about1.2 larger than in the V data. The harmonics of the mainfrequency as well as the triplet structure around the mainfrequency and its harmonics are detectable up to high or-der, especially in the B data (15th order). Also the Blazhko modulation is a factor of 1.2 larger in the B than in the V data. • After subtracting the fit including triplet components,we find clear evidence of quintuplet frequencies in thedata. We found several clearly significant peaks located at kf ± f B frequency values. These frequencies emerged mostclearly from the B data set in which the modulation compo-nents have higher amplitudes, but were also retrieved fromthe V data. Our data thus clearly show the existence of quin-tuplet components in SS For, as were also found by Hurtaet al. (2008) in RV UMa and by Jurcsik et al. (2008) inMW Lyr. Quintuplet components are predicted by the mag-netic model for explaining the Blazhko effect (Shibahashi& Takata 1995), but so far there has been no convincing,unambiguous positive detection of a magnetic field in RRLyrae stars. • We also find evidence for a so-called septuplet compo-nent in our data (at 6 f + 3 f B ). Multiplet components oforder j (cid:62) kf ± jf B have also been found by Jurcsik etal. (2008) in their extensive data set of MW Lyr. They haveyet to be explained by the models for the Blazhko effect. Un-interrupted satellite data of Blazhko stars, covering severalBlazhko cycles, such as delivered by the COROT satellite(Baglin 2007), will undoubtedly shed new light upon the ex-istence of higher-order multiplet structures in Blazhko stars.We may expect to find many more in high-quality data sets. • A subdivision of the data into 10 Blazhko phase in-tervals shows the variations of the light curve over theBlazhko cycle. As all the Blazhko phase intervals in V havesufficient coverage, we calculated the Fourier parametersand plotted their variations. Observed values at differentphases of the Blazhko cycle may be useful for confronta-tion with non-linear convective models such as those devel-oped by Feuchtinger (1999). It would be worthwhile to findout whether the observed variations in the Fourier parame-ters can be reproduced theoretically for a star with constantmass and metallicity. • Application of the empirical P − φ − [Fe/H] relationdeveloped by Jurcsik & Kov´acs yields a good result for theaverage light curve derived from all our data. This strength-ens the assumption that the average light curve of Blazhkostars can be employed in the empirical formula calibratedby Jurcsik & Kov´acs on stricly periodic stars. However, theaverage light curve of a Blazhko star can only be determinedaccurately if the Blazhko cycle is sufficiently covered by theavailable data. • From a study of the light curve shape at differentBlazhko phases we clearly see that the strong variationsaround minimum light in SS For are related to the positionand strength of the bump. A detailed study of the bumpbehaviour over the Blazhko cycle, in photometric as wellas spectroscopic data, may shed new light upon the under-standing of the Blazhko effect.Theoretical efforts to revise or expand the existing modelsfor the Blazhko effect would be worthwhile, as well as theexploration of alternative explanations. c (cid:13)000
05 d. • The amplitudes in the B data are a factor of about1.2 larger than in the V data. The harmonics of the mainfrequency as well as the triplet structure around the mainfrequency and its harmonics are detectable up to high or-der, especially in the B data (15th order). Also the Blazhko modulation is a factor of 1.2 larger in the B than in the V data. • After subtracting the fit including triplet components,we find clear evidence of quintuplet frequencies in thedata. We found several clearly significant peaks located at kf ± f B frequency values. These frequencies emerged mostclearly from the B data set in which the modulation compo-nents have higher amplitudes, but were also retrieved fromthe V data. Our data thus clearly show the existence of quin-tuplet components in SS For, as were also found by Hurtaet al. (2008) in RV UMa and by Jurcsik et al. (2008) inMW Lyr. Quintuplet components are predicted by the mag-netic model for explaining the Blazhko effect (Shibahashi& Takata 1995), but so far there has been no convincing,unambiguous positive detection of a magnetic field in RRLyrae stars. • We also find evidence for a so-called septuplet compo-nent in our data (at 6 f + 3 f B ). Multiplet components oforder j (cid:62) kf ± jf B have also been found by Jurcsik etal. (2008) in their extensive data set of MW Lyr. They haveyet to be explained by the models for the Blazhko effect. Un-interrupted satellite data of Blazhko stars, covering severalBlazhko cycles, such as delivered by the COROT satellite(Baglin 2007), will undoubtedly shed new light upon the ex-istence of higher-order multiplet structures in Blazhko stars.We may expect to find many more in high-quality data sets. • A subdivision of the data into 10 Blazhko phase in-tervals shows the variations of the light curve over theBlazhko cycle. As all the Blazhko phase intervals in V havesufficient coverage, we calculated the Fourier parametersand plotted their variations. Observed values at differentphases of the Blazhko cycle may be useful for confronta-tion with non-linear convective models such as those devel-oped by Feuchtinger (1999). It would be worthwhile to findout whether the observed variations in the Fourier parame-ters can be reproduced theoretically for a star with constantmass and metallicity. • Application of the empirical P − φ − [Fe/H] relationdeveloped by Jurcsik & Kov´acs yields a good result for theaverage light curve derived from all our data. This strength-ens the assumption that the average light curve of Blazhkostars can be employed in the empirical formula calibratedby Jurcsik & Kov´acs on stricly periodic stars. However, theaverage light curve of a Blazhko star can only be determinedaccurately if the Blazhko cycle is sufficiently covered by theavailable data. • From a study of the light curve shape at differentBlazhko phases we clearly see that the strong variationsaround minimum light in SS For are related to the positionand strength of the bump. A detailed study of the bumpbehaviour over the Blazhko cycle, in photometric as wellas spectroscopic data, may shed new light upon the under-standing of the Blazhko effect.Theoretical efforts to revise or expand the existing modelsfor the Blazhko effect would be worthwhile, as well as theexploration of alternative explanations. c (cid:13)000 , 000–000 K. Kolenberg, et al.
Figure 14. B light data (crosses) in different 0.1 phase intervals of the Blazhko cycle (34.94 ± ACKNOWLEDGMENTS
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