Mon. Not. R. Astron. Soc., () Printed 23 April 28 (MN LATEX style file v2.2) Period-Colour and Amplitude-Colour Relations for RR Lyraes Stars in M3 and M5 S. M. Kanbur, G. Feiden, W. Spratt, C. Ngeow 2, R. Szabo 3 and R. Buchler 3 Department of Physics, State University of New York at Oswego, Oswego, NY 326, USA 2 Department of Astronomy, University of Illinois, Urbana-Champaign, IL 68, USA 3 Department of Physics, University of Florida, Gainesville, FL 326, USA Accepted 27 month day. Received 27 month day; in original form 27 April 5 ABSTRACT In this paper, we analyze period-colour (PC) and amplitude-colour (AC) relations for RR Lyrae stars both theoretically and observationally. By analyzing M3 data that for RRab fundamental mode stars, we find the period-colour relation using both (B V ) and (V I) colours is flat at minimum light and that amplitude-colour relations are consistent with a simple application of the Stefan-Boltzmann law such that higher amplitude stars are driven to a bluer colour at maximum light. First overtone RRc stars do not exhibit a flat PC relation at minimum light. However for both types of stars, the relation at mean light always lies in between that at maximum and mean light. We also find that period-amplitude relation which is thought to differentiate between Oosterhoff type clusters I and II can be expressed as a period-colour and amplitudecolour relation at a given phase. Since these latter two relations are a function of phase, the implication is that a deeper understanding of the nature of the Oosterhoff dichotomy will be obtained by a detailed study, both theoretically and observationally, of the properties of these stars as a function of phase. We also compute theoretical models for OoI and OoII type clusters and compare our theoretical period-colour and amplitude-colour relations with that from observations. The theoretical models do follow PC and AC relations of the same form as the observations thus confirming previous work. The models suggest some new ways to calculate extinction for RR Lyraes using PC relations at minimum light for long period (log P.2) RRab stars or using AC relations. We also carry out a Principal Component Analysis of RR Lyrae stars in the LMC and show that the first Principal Component, which carries about 7% of the light curve structure is strongly correlated with amplitude. Thus comparing models and theory on period-amplitude diagrams is a reasonable way to compare theoretical and observed light curves: this is Bailey type diagrams work. Key words: RR Lyraes Stars: fundamental parameters INTRODUCTION?hereafter KF)kanbur and fernando25 studied RRab stars in the Large Magellanic Cloud and presented period-colour (PC) relations at maximum, mean and minimum V band light. Using a colour defined by the MACHO V and R passbands, they found a flat PC relation at minimum light, though with considerable scatter, and a definite relation at maximum light with non-zero slope. The PC relation at mean light, the relation that is usually discussed in the literature, had a slope intermediate between the two extremes at maximum and minimum light. KF also provided evidence Email: kanbur@oswego.edu that, at least in their sample of about 8 RRab stars in the LMC, the PC relation slope varied significantly with phase. The Oosterhoff dichotomy separates globular clusters into two types, based on the properties of their RR Lyrae stars. An excellent review is given in Cacciari et al (25, hereafter C5). The main difference between the two groups is the mean period of their RRab stars: OoI and OoII clusters have a mean period of.65 and.55 days respectively. C5 also describe other properties that differentiate the two types of clusters. One of these is the period-amplitude (PA) relation. C5 contend that the PA relation in OoI and OoII is distinct and independent of metallicity of cluster in a given Oosterhoff group. They also find evidence that evolved RRab stars in OoI clusters follow similar PA relations to c RAS
2 Kanbur et al. those obeyed by OoII clusters. Finally they find that these statements are followed by both RRc and RRab stars. If the PA relation is written as V A = a + b log P, () where V A is the V band amplitude, then it can be shown that, V A = a + c(b V ) c(b V ) + b log P, (2) or V A = a + d(v I) d(v I) + blog P. (3) In these equations the colour index, whether (B V ) or (V I) can be taken at any phase. Note that for different phases, the values of c and d may be different. Because PA relations are distinct in the two Oosterhoff clusters, these equations imply the existence of PC relations which can also distinguish between the two Oosterhoff type clusters. Also, Kanbur and Fernando (25), Kanbur and Phillips (996) by applying the Stefan-Boltzmann law at maximum and minimum V band light, then and V min V max = b[(v I) max (V I) min], (4) V max V min = b [(B V ) max (B V ) min], (5) where b and b are constants. Hence if either (V I) or (B V ) follow a flat or flatter relation with period at maximum or minimum light, then there will be a relation between V band amplitude and colour at minimum or maximum light - that is an amplitude-colour (AC) relation. The aim of this paper is to start an investigation into PC/AC relations as a function of both phase and Oosterhoff type with a view to providing an understanding of equation () and the findings of C5 through equations (2) and (3). We use data from two globular clusters in this study: M3 and M5. These are traditionally regarded as the archetypal OoI and OoII clusters respectively. For M3 we use data from Hartmann et al (24, hereafter H4) and for M5 we use data from Silbermann and Smith (995, hereafter SS). 2 THE DATA The photometric data used in this study are described more fully in H4 and SS and consist of BV I band data for RR Lyraes in both M3 and M5, respectively. We only consider the fundamental mode (RRab, classified as RR in H4) and first overtone (RRc, classified as RR in H4) RR lyraes as given in H4 and SS in our study. RR Lyraes without the photometric data in all three BV I bands and/or the number of data points per light curves in either bands is less than are omitted for further analysis. For the rest of the RR Lyraes in both sets of data were Fourier analyzed according to the expression of: k=n X X = A + A k cos(kωt + φ k ), (4) k= where n is the order of the fit and X is the waveband, either B, V or I. The order of fit, n, is allowed to vary.5 -.6 -.4 -.2 Figure. The period-amplitude relation for the RR Lyraes in M3 and M5. between 2 to 6 and the best-fit light curve is chosen by visual inspection. Occasionally we improve the fits by using the simulated annealing method described in Ngeow et al. (23). One of the requirements of the data is that for a given star, both B and V, or equivalently, V and I band data exist of sufficient quality to produce a colour curve. Therefore we eliminate RR Lyraes that show abnormal light curves, light curves with unsatisfactory fits and those stars that exihibit obvious Blazhko effect. This left 4 stars (8 for RR and 23 for RR) and 4 stars (3 for RRab and for RRc) in M3 and M5, respectively. The Fourier fits in each waveband were used to compute colours at different phases. The three phases of interest are maximum, mean and minimum V band light and the colour when the star s V band luminosity approaches it mean value (Kanbur and Ngeow 24). 3 RESULTS Figures -5 present our results in the form of PC and AC diagrams at the three different phases, max, mean and minimum V band light, using the two colours (B V ) and (V I). Solid and open squares represent M3 and M5 stars respectively. and open squares denote M3 and M5 data respectively. Note these diagrams have not been corrected for extinction. Tables -2 present quantitative results for M3 from our data analysis which can be used for future reference. We concentrate on M3 because the M5 data is not sufficient to quantitatively compare with M3. Thus tables -2 use subset of the M3 data, some stars, for which the Fourier decompositions are acceptable in the sense that there are no unphysical bumps or wiggles due to poor phase coverage. These tables display the results of linear regressions, Y = a + bx, Y, X the dependent and independent variable respectively for a number of different pairs (X, Y ). We note that all the plots display two groups. These are clearly the first and fundamental mode RR Lyraes at c RAS, MNRAS,
PC & AC Relations for RR Lyrae 3.8.8.6.6.4.4.2.2.8.8.6.6.4.4.2.2 -.6 -.4 -.2.5 Figure 2. The period-colour (left panel) and amplitude-colour (right panel) relations for the RR Lyraes in M3 and M5 at maximum light..8.8.6.6.4.4.2.2.8.8.6.6.4.4.2.2 -.6 -.4 -.2.5 Figure 3. The period-colour (left panel) and amplitude-colour (right panel) relations for the RR Lyraes in M3 and M5 at mean light. Table. The period-colour relations for RR Lyraes in M3. (B V ) Colour (V I) Colour Phase Slope Zero-point Slope Zero-point >.4 Maximun.765 ±.56.46 ±.4.78 ±.273.535 ±.7 Mean.49 ±.28.5 ±.33.44 ±.89.62 ±.49 Minimum.22 ±.5.456 ±.39.2 ±.27.69 ±.54 <.4 Maximun.585 ±.325.55 ±.62.662 ±.539.572 ±.269 Mean.666 ±.34.589 ±.7.522 ±.482.6 ±.24 Minimum.858 ±.443.77 ±.22.53 ±.49.68 ±.245 c RAS, MNRAS,
4 Kanbur et al..8.8.6.6.4.4.2.2.8.8.6.6.4.4.2.2 -.6 -.4 -.2.5 Figure 4. The period-colour (left panel) and amplitude-colour (right panel) relations for the RR Lyraes in M3 and M5 at minimum light. Table 2. The amplitude-colour relations for RR Lyraes in M3. (B V ) Colour (V I) Colour Phase Slope Zero-point Slope Zero-point >.4 Maximun.225 ±.9.48 ±.7.45 ±.29.57 ±.26 Mean.6 ±.24.427 ±.2.68 ±.34.57 ±.29 Minimum.37 ±.27.43 ±.23.32 ±.37.67 ±.32 <.4 Maximun.496 ±.47.424 ±.63.765 ±.238.567 ±.2 Mean.434 ±.7.44 ±.73.545 ±.23.572 ±.99 Minimum.533 ±.225.56 ±.97.389 ±.25.595 ±.8 short (log P.4) and long (log P.4) respectively. For this reason many of the regression results in table (xx) are separated into two groups. We note that for M5 the majority of the colour data are for first overtone stars and that there are insufficient fundamental mode M5 stars for a proper comparison, at least for this mode of oscillation. It is clear that, in terms of PC and AC relations, the first overtone stars in M5 are different to those in M3, because the M5 stars have both longer periods and redder colours. This is easily apparent in the PC relation at maximum rather than mean light. The PC relation at minimum light for B V and V I colors is flat. The scatter at minimum light around a flat PC relation is about.4 mags for both colours. At maximum light, for both colours we see a clear relation such that longer period stars have redder colours at maximum V band light. It is clear that the behaviour at mean light for fundamental model stars is intermediate between that at maximum or minimum V band light. The tables show clearly that the regressions at mean light are usually in between the corresponding values at minimum of maximum light. Hence a deeper understanding of the PC behaviour at mean light can be obtained by studying these PC relations as a function of phase. There are only 3 fundamental mode RR Lyraes from M5 which have sufficient data to obtain reliable colours at these three phases. It is difficult to conclude anything though they all lie above the PC relations for M3 (ie. at redder colours) at the same phase. We see that for the overtone (or shorter period stars), the periods of the M5 stars are generally longer than for M3. This is consistent with the idea that OoII stars are more evolved than those in OoI clusters. 4 THEORETICAL MODELS We use the pulsation codes of Yecko et al (999) and Kollath et al (22) and Szabo et al (24) to compute a large grid of full amplitude RRab models with the following composition: appropriate for OoI and OoII clusters respectively: X =.75, Y = xxx,z =.,. Each model is specified by a mass, luminoisity, effective temperature and composition with.3m < M <.9M, L < L < L, 58K < T eff < 7K. After full amplitude pulsation was reached, the theoretical variations of the photospheric c RAS, MNRAS,
PC & AC Relations for RR Lyrae 5.5 Z=..5 Z=..5.5 Z=..5.5 Z=. Distance to photosphere (min).3..5.3 Z=. Distance to photosphere (min).3..5.3 Z=. Z=. Z=... -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -...5 Z=..5 Z=..5.5 Z=..5.5 Z=. Distance to photosphere (min).3..5.3 Z=. Distance to photosphere (min).3..5.3 Z=. Z=. Z=... -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -.. Figure 5. The period-amplitude relation from theoretical models. temperature were converted to V I colours using Kurucz model atmospheres. This resulted in PC and AC relations as a function of phase, in particular at maximum, mean and minimum light. 4. Model Results The model grid parameters and the resulting linear periods are presented in Tables 3 and 4. Figure 5 displays the theoretical PA relation. We see that with these models the amplitudes tend to be a bit smaller than those observed. Of particular interest are figures 5-7 which depict the PC/AC relations at maximum and minimum light. At minimum light, we clearly see a much flatter relation extending the work of Kanbur (995), Kanbur and Phillips (996) on the flat PC relations of RRab stars at minimum light. For periods greater than log P >.2, the scatter in the theoretical PC relation is clearly smaller. The flat PC relation at minimum light has been used to estimate reddenings for globular clusters (Sturch xxxx, Clementini et al xxxx, Guldenshu et al xxxx, Sarajeni et al 26). The left panel of Figure 7 implies that considering only stars with log P >.2 would lead to a more accurate reddening estimate since the scatter is considerably reduced. The observational results in the right panel of figure 4 do also tend to support this. The AC relations are what is expected from Kanbur and Phillips (996). At maximum light, higher amplitude stars are driven to hotter temperatures and hence bluer V I colours because the colour at minimum light does not vary much with period. Kanbur and Fernando (25) provide simple arguments using the Stefan Boltzmann law to justify this. Again the right panels of figure 7 imply that well calibrated AC relations at maximum/minimum light can be used to estimate extinction because the scatter of such relations is small. In comparing theoretical and observed PC/AC relations, the models tend to have a smaller amplitude and hence cooler colors than the observations. However there is a match between observations and theory Figure 8. SHASHI: fill in the caption here when considering the qualitative form of the PC/AC relations. Using the definitions given in KNB and KN, figure 8 displays the distance to the photosphere from the hydrogen ionization front (HIF) at minimum light. We see that the distance is essentially independent of period: for this entire period range, the photosphere is located at the base of the HIF. In this situation, because the photospheric densities are low, the temperature at which hydrogen ionizes and hence the temperature of the photosphere is essentially independent of density and hence of global stellar parameters. Thus RRab stars exhibit flattish PC relations at minimum light. 5 PRINCIPAL COMPONENT ANALYSIS Kanbur et al (22), Kanbur and Mariani (24, hereafter KM), Tanvir et al (25), Leonard et al (24) developed the idea of using Principal Component Analysis (PCA) to analyze the structure of Cepheid and RR Lyrae light curves. There are a couple of advantages over the tradditional Fourier method. Firstly, it is more efficient: whereas an eight order fit with 6 parameters are typicallyr equired to describe the structure of a light curve, the same structure can be described with 4 PCA parameters. In fact one PCA parameter describes as much as 8% of the variation in light curve structure (KM). Secondly, by construction, the PCA Coefficients are othogonal to each other and hence carry independent information about light curve structure. In contrast, gigher Fourier amplitudes are often correlated with each other. The left panel of figure 9 prtrays the first principal component derived from a PCA analysis of about 78 stars from the MACHO Project RR Lyraes studied in Kanbur and Fernando (24). We clearly see a feather like region in the PC plot where the majority of stars lie. We find that amplitude generally correlates with the PC value: larger/lower amplitude stars are located in the lower/upper parts of this feather region respectively. Since our previous work has suggested that amplitude correlates with photospheric colour at maximum light in the sense that higher c RAS, MNRAS,
6 Kanbur et al. Table 3. Grid model parameters for z =. M log L T eff P (days) M log L T eff P (days).55.5 6.663.65.5 6.5572.55.5 6.587.65.5 6.5255.55.5 62.54857.65.5 62.499.55.5 63.5863.65.5 63.46546.55.5 64.4998.65.5 64.44.55.5 65.46537.65.5 65.4834.55.5 66.4463.65.5 66.39727.55.5 67.4954.65.5 67.37766.55.5 68.39898.65.5 68.35936.55.5 69.37979.65.5 69.34228.55.5 7.3687.65.5 7.3263.55.6 6.7487.65.6 6.66835.55.6 6.7472.65.6 6.6325.55.6 62.66484.65.6 62.59524.55.6 63.6283.65.6 63.56296.55.6 64.59428.65.6 64.5332.55.6 65.56297.65.6 65.5548.55.6 66.53396.65.6 66.4798.55.6 67.57.65.6 67.45595.55.6 68.4895.65.6 68.4337.55.6 69.45862.65.6 69.4294.55.6 7.4368.65.6 7.39353.55.72 6.92725.65.72 6.82659.55.72 6.87256.65.72 6.77889.55.72 62.82245.65.72 62.7357.55.72 63.77638.65.72 63.69473.55.72 64.73394.65.72 64.65749.55.72 65.69479.65.72 65.6234.55.72 66.65856.65.72 66.599.55.72 67.625.65.72 67.564.55.72 68.59382.65.72 68.53377.55.72 69.56477.65.72 69.582.55.72 7.5377.65.72 7.48397 amplitude stars are driven to hotter temperatures and hence bluer colours at maximum light, we find that the PC coefficient correlates with the photospheric colour at maximum light in the sense that the bluer stars at maximum light are found in the lower parts of this feather region. This links the light curve structure with a property of the pulsating envelope. The right panel of figure 9 is the same diagram but for the RR Lyrae stars in M3. We see that this diagram has broadly the same shape as the PCA diagram for the MACHO project RR Lyraes, the difference being the significantly lower number of stars. Period-amplitude diagrams have been traditionally used to classify RR Lyraes into Bailey type ab or c. This has generally translated into fundamental and first overtona stars, respectively, which both have very different light curve structure. We see now why period-amplitude diagrams work, at least as a first approximation, in this classification. The PC coefficient describes more than 8% of the light curve variation in RR Lyrae light curves. This coeffficient is correlated with amplitude. Hence a period-amplitude diagram is, to first level of approximation, a period-light curve structure diagram. 6 CONCLUSION AND DISCUSSION We have found that for globular cluster RRab stars in M3 the PC and AC relations follow the framework outlined in KF. This is true for both colours (B V ) and (V I). Specifically the PC relation at minimum V band light is flat and the colour at maximum light is bluer when the amplitude is higher. These results can be explained by a simple application of the Stefan Boltzmann law applied at maximum and minimum light and the interaction of the photosphere and hydrogen ionization front (Kanbur 995, Kanbur and Phillips 996), though a newer series of pulsation of models is currently being constructed to critically investigate this. RRc stars in both M3 and M5 do not exhibit a flat PC relation at minimum light and more theoretical woork is needed to understand why. However, the M5 first overtone RR Lyraes have longer periods than their counterparts in M3 and also different PC and AC relations. For both fundamental and first overtone modes, the PC and AC relations, which are equivalent to the PA relations, are a strong function of phase and the mean light relations are always intermediated between those at maximum and minimum light. Since the PA relations have been used to differentiate between Oosterhoff groups (Cacciari et al 25) our main conclusion is that studying the properties of RR c RAS, MNRAS,
PC & AC Relations for RR Lyrae 7 Table 4. Grid model parameters for z =. M log L T eff P (days) M log L T eff P (days).55.5 58.69.55.5 6.6963.65.5 6.54447.55.5 6.57423.65.5 6.5335.55.5 62.5476.65.5 62.48479.55.5 63.588.65.5 63.45846.55.5 64.4843.65.5 64.4344.55.5 65.45879.65.5 65.46.55.5 66.4354.65.5 66.3968.55.5 67.437.65.5 67.3722.55.5 68.39274.65.5 68.3539.55.5 69.37368.65.5 69.3366.55.5 7.35589.65.5 7.3234.55.6 58.84.55.6 6.7469.65.6 6.6633.55.6 6.6972.65.6 6.62227.55.6 62.65735.65.6 62.58732.55.6 63.6267.65.6 63.5553.55.6 64.58686.65.6 64.5254.55.6 65.5556.65.6 65.49789.55.6 66.5267.65.6 66.47236.55.6 67.49988.65.6 67.44864.55.6 68.47493.65.6 68.42655.55.6 69.456.65.6 69.4596.55.6 7.433.65.6 7.3867.55.72 58.44.55.72 6.99.65.72 6.8768.55.72 6.86426.65.72 6.76998.55.72 62.849.65.72 62.72622.55.72 63.768.65.72 63.68594.55.72 64.72562.65.72 64.64877.55.72 65.68653.65.72 65.644.55.72 66.6537.65.72 66.58262.55.72 67.6688.65.72 67.5538.55.72 68.58584.65.72 68.5256.55.72 69.55693.65.72 69.5.55.72 7.52998.65.72 7.476 Lyraes as a function of phase can increase our insight into the Oosterhoff dichotomy and hence help to improve the RR Lyrae age and distance scales. ACKNOWLEDGMENTS We thank Carla Cacciari for helpful discussions and an anonymous referee for useful suggestions. CN acknowledges support from NSF award OPP-362 and a University of Illinois seed funding award to the Dark Energy Survey. REFERENCES Kanbur, S. M. & Fernando, I., 25, MNRAS, 359, L5 c RAS, MNRAS,
8 Kanbur et al..8 Z=..8 Z=..6.4.6.4.2.2.8 Z=..8 Z=..6.4.6.4.2.2 -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -...8 Z=..8 Z=..6.4.6.4.2.2.8 Z=..8 Z=..6.4.6.4.2.2 -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -...8.8.6.4.6.4.2.2 Z=. Z=..8.8.6.4.6.4.2.2 Z=. Z=..5.5.5.5.8.8.6.4.6.4.2.2 Z=. Z=..8.8.6.4.6.4.2.2 Z=. Z=..5.5.5.5 Figure 6. The period-colour (left panel) and amplitude-colour (right panel) relations from theoretical models at maximum light. c RAS, MNRAS,
PC & AC Relations for RR Lyrae 9.8.8.6.6.4.4.2.2 Z=. Z=..8.8.6.6.4.4.2.2 Z=. Z=. -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -...8.8.6.6.4.4.2.2 Z=. Z=..8.8.6.6.4.4.2.2 Z=. Z=. -.5 -.4 -.3 -.2 -.. -.5 -.4 -.3 -.2 -...8.8.6.6.4.4.2.2 Z=. Z=..8.8.6.6.4.4.2.2 Z=. Z=..5.5.5.5.8.8.6.6.4.4.2.2 Z=. Z=..8.8.6.6.4.4.2.2 Z=. Z=..5.5.5.5 Figure 7. The period-colour (left panel) and amplitude-colour (right panel) relations from theoretical models at minimum light. c RAS, MNRAS,
Kanbur et al..5.5 -.5 -.5 -.6 -.4 -.2 -.6 -.4 -.2 Figure 9. P C coefficient against log P for MACHO Project LMC RR Lyraes (left panel) and RR Lyraes in M3 & M5 (right panel) in the V band. c RAS, MNRAS,