Abstract

We present period–luminosity relations for more than 3200 red variable stars in the Small Magellanic Cloud observed in the second phase of the Optical Gravitational Lensing Experiment (OGLE II). Periods of multiply periodic light curve solutions combined with the single epoch 2MASS JHKS magnitudes, reveal very similar distributions to those for the Large Magellanic Cloud in Paper I. The main features include four pulsating asymptotic giant branch (AGB) ridges, three distinct short period sequences below the tip of the red giant branch (RGB) and two long period sequences of ambiguous origin. We derive a relative distance modulus of the Clouds from the period–luminosity distributions for all stars of Δμ0 = 0.44 mag, which is in good agreement with recent independent results. The tip of the RGB shows a colour and metallicity dependence that is in excellent agreement with the empirical results for globular clusters. We conclude that most variable stars below the tip of the red giant branch are indeed RGB stars.

INTRODUCTION

The period–luminosity (P–L) relations of long period variables on the asymptotic giant branch (AGB) are well documented for the Large Magellanic Cloud. Observations have revealed not only that Mira variables follow a tight near infrared P–L relation (Glass & Lloyd Evans 1981, 2003; Feast et al. 1989), but also that semiregular variables pulsating in overtone modes follow distinct and well defined P–L relations (Wood et al. 1999; Wood 2000). In recent years, there has been strong interest in pulsating red giants, which is reflected in the increasing number of independent analyses of large photometric data bases (Cioni et al. 2001, 2003; Noda et al. 2002; Lebzelter, Schultheis & Melchior 2002; Glass & Schultheis 2003; Kiss & Bedding 2003; Ita et al. 2003; Groenewegen 2004).

An interesting by product was the recognition of a distinct and very numerous group of red variables below the tip of the red giant branch (TRGB). First, Ita et al. (2002) noted that the luminosity function (LF) of red giant variable stars in the Large Magellanic Cloud (LMC) shows a sharp feature at the expected brightness of the TRGB. Earlier, Alves et al. (1998) and Wood (2000) suggested that thermally pulsing AGB stars can mimic a two peaked luminosity function, and thus the LF shape alone is not sufficient to establish the existence of TRGB variables. However, we have shown in Paper I (Kiss & Bedding 2003) that the LMC period–luminosity relations for stars above and below the TRGB show a relative period shift that is consistent with the evolutionary temperature difference between RGB and AGB stars. This result has since been confirmed by Ita et al. (2003). The possibility of observable pulsations in RGB stars therefore seems to be well established, providing a new area for asteroseismological considerations. In this Letter we provide further evidence for extragalactic RGB variables through a comparative analysis of red variables in the Small Magellanic Cloud (SMC).

Red giant pulsators in the SMC have a far less extensive literature than those of the LMC. The largest pre microlensing data sets for the SMC were published by Wood, Bessell & Fox (1981), Lloyd Evans, Glass & Catchpole (1988) and Sebo & Wood (1994), all consisting of a few dozen variables at most. Microlensing surveys (MACHO, EROS, OGLE) have changed the situation by discovering thousands of red variables. Based on the extensive new data sets, it has become possible to compare the Magellanic Clouds and the Galactic bulge in search for metallicity effects on red giant pulsations and P–L relations. Cioni et al. (2003) examined MACHO light curves of 458 long period variables (LPVs) in the SMC that were detected by the Infrared Space Observatory. They found very similar P–L sequences to those in the LMC and confirmed the relative overabundance of carbon rich Mira stars (as recognized first by Lloyd Evans et al. 1988). The most extensive analysis to date is that of Ita et al. (2003), who determined the dominant period for 2927 variables in the SMC using OGLE II data and constructed P–L relations using single epoch SIRIUS JHK survey data. Besides finding the same distributions as those in Paper I, Ita et al. (2003) derived zero point differences of the P–L relations of about 0.1 mag. Some differences were also noted recently by Cioni et al. (2003) for the Magellanic Clouds and by Glass & Schultheis (2003) for the Galactic bulge and the Magellanic Clouds. The latter investigators found both slightly different slopes and different absolute magnitude ranges for the overtone P–L sequences. These interesting results depend on the assumed distances to each of the different P–L data sets. Here we concentrate on the multiwavelength luminosity functions to sort out the distance issue, and examine the implications concerning the RGB pulsations.

Data Analysis

The basis of our analysis is the catalogue of OGLE II (Udalski, Kubiak & Szymanski 1997) data spanning four years from 1997 to 2000 (Zebrun et al. 2001). Of the total 7 deg2 of the OGLE Magellanic Cloud fields, about 2.5 deg2 were observed in the SMC. The observations were analysed by the OGLE team using a modification of the Difference Image Analysis (Alard & Lupton 1998; Wozniak 2000), which is the most advanced method available for detecting small variations in crowded fields (see, e.g. Bonanos & Stanek 2003).

Typical light curves in the SMC consist of 280 points distributed over ∼1100 d. Compared with the LMC, where individual data sets covered slightly more than 1200 d in 400 points, this means a reduced accuracy in period determination. We consider our periods reliable up to about 400–500 d, for which at least two cycles were observed.

The data were analysed in the same manner as in Paper I. To summarize, we performed the following steps.

Our sample is slightly larger than that of Ita et al. (2003) because of the different selection criteria. A more relevant difference is that we calculated a set of periods for every star, while Ita et al. (2003) determined only one period per star. We found multiple periodicity for the majority of stars (82 per cent of the full sample), which can, therefore, be identified as common behaviour in red giants. There must be, of course, a fraction of spurious or physically unrelated periods, which need a more sophisticated method for identification and exclusion from future analyses.

Discussion

We present the resulting P–L diagram in Fig. 1. At first glance, the shape and extension of distributions are very similar to those we found for the LMC.

Figure 1.

P–L relations for red giants in the SMC (10 009 periods for 3260 stars). This figure is available in colour in the online version of the journal on Synergy.

Figure 1.

P–L relations for red giants in the SMC (10 009 periods for 3260 stars). This figure is available in colour in the online version of the journal on Synergy.

  • There is a noticeable drop of point density at Ks≈ 12.70 mag, located close to the tip of the RGB (Cioni et al. 2000).

  • Above the tip, there are four pulsating AGB sequences, which in the LMC were identified by Wood (2000) as stars pulsating in the fundamental (F), first (1O), second (2O) and third (3O) overtone modes (he labelled them as sequences C, B and A which, however, does not account for the fact that there are four ridges clearly distinguishable in the OGLE II data).

  • Below the tip, there are three distinct sequences in the short period range (P < 60 d), of which R2 and R3 are continuations of 2O and 3O; there might be a slight horizontal shift, similar to that found in the LMC (Paper I), but the ridges are less well defined, owing to the smaller number of stars, so the possible period shift between 3O –R3 and between 2O –R2 is barely measurable. The sequence R1, just as for the LMC, lies somewhere between the extrapolated 1O and F ridges.

  • For longer periods, there are two sequences, L1 and L2, in the same relative position to the other ridges as in the LMC. Their interpretation is ambiguous; Wood (2000) suggested L1 (=E in his paper) contains eclipsing binary variables, while L2 (=D) may consist of pulsating stars with unknown excitation mechanism or non spherical stars with rotationally induced variations (Olivier & Wood 2003; Wood 2004).

Although multiperiodicity is a common feature, there is a remarkable difference above and below the tip of the RGB: for KS < 12.7 mag, 95 per cent, while for KS > 12.7, only 62 per cent of the light curves resulted in multiple periods. This, however, only reflects the fact that amplitudes below the TRGB strongly decreases towards fainter magnitudes (see Section 3.3), thus the fixed 5 mmag amplitude threshold may be too high for most of the periods (if they exist). In addition, we stress there is not any noticeable change between the resulting P–L distributions when plotting every star just once using the strongest period or using all periods with amplitudes exceeding a certain threshold (or even plotting every star just once with the second strongest period).

The relative distance modulus of the Clouds

In the case of such complex structures as the P–L distributions of red variables in the Magellanic Clouds, it is difficult to measure the vertical distance between the P–L ridges directly. It has been a more common approach to assume a relative distance modulus and overplot the relations with the purpose of comparison, either in a simplified functional form (see fig. 11 in Glass & Schultheis 2003) or showing the full observational data (fig. 10 in Ita et al. 2003). Here we follow a different approach to show that, assuming a negligible mean zero point difference for the set of relations, one can get consistent relative distance moduli and extinctions of the Clouds in all four photometric bands (IJHKS), which will be used later to characterize the dips of the luminosity functions.

Instead of a visual examination of overplotted data, we calculated the cross correlation functions of smoothed images of the P–L ridges (being the sharpest in KS and the least defined in I). For this, we converted the scatter diagrams (Fig. 1 and fig. 1 in Paper I) to 500 × 500 pixel images as follows. The x axis covered log P between 1.2 and 3.1 (3.8 × 10−3 dex pixel−1), while the y axis covered 5 magnitudes, where the exact range depended on the studied band (e.g. from 9 mag to 14 mag in KS). This way one pixel in vertical direction corresponds to 0.01 mag. Every pixel ‘intensity’ value was set to the total number of stars in a 21 × 21 pixel mask, centred on the actual pixel. This masking worked as a boxcar smoothing procedure, which produced clear images of the P–L ridges. The cross correlation functions of the IJHKS P–L images were calculated as functions of Δμ, and their maxima showed the vertical shifts that give the best overlap of the images. In order to minimize the effects of spurious periods, we kept only stars with log P < 2.6 (i.e. P < 1 yr); in other words, we used only 360 × 500 pixel sub images.

The resulting shifts are presented in the second column of Table 1. There is a strong wavelength dependence, which can be explained by the differential reddening. The average E(BV) for the LMC and the SMC are 0.15 mag and 0.065 mag, respectively (Westerlund 1997), which can be translated to ΔAX differential extinction by adopting RV = 3.1 and the relative extinction coefficients (AX/AV) in Schlegel, Finkbeiner & Davis (1998).

Table 1.

Relative distance moduli and extinctions. Δμ is the mean vertical distance between the P–L images;ΔAX refers to the differential extinction in the X band, while Δμ0 is the dereddened relative distance modulus.ΔE(B-V) = 0.085 (Westerlund 1997) and ΔAV = 0.26 mag were assumed. The uncertainty in Δμ is about ±0.05 mag.

Table 1.

Relative distance moduli and extinctions. Δμ is the mean vertical distance between the P–L images;ΔAX refers to the differential extinction in the X band, while Δμ0 is the dereddened relative distance modulus.ΔE(B-V) = 0.085 (Westerlund 1997) and ΔAV = 0.26 mag were assumed. The uncertainty in Δμ is about ±0.05 mag.

As can be seen in Table 1, the dereddened relative distance moduli agree well within the error bars (∼±0.05 mag in every band). Similar agreement exists for recently published results; for instance, Cioni et al. (2000) determined Δμ= 0.44 mag. On the other hand, Ita et al. (2003) adopted Δμ= 0.49 mag, while Glass & Schultheis (2003) used 0.50 mag. In the rest of the paper we adopt 0.44 mag with an estimated ±0.03 mag random error.

The luminosity functions and the tip of the RGB

In Fig. 2 we plot luminosity functions in all four bands of the OGLE II+2MASS data base: the thin solid line shows the LMC data after correction for the different survey areas (4.5 deg2 for the LMC versus 2.5 deg2 for the SMC); the thick solid line denotes the SMC LF.

Figure 2.

Stellar magnitude distributions for OGLE II variables in the SMC (thick red line) and in the LMC (thin black line), normalized by the survey area. This figure is available in colour in the online version of the journal on Synergy.

Figure 2.

Stellar magnitude distributions for OGLE II variables in the SMC (thick red line) and in the LMC (thin black line), normalized by the survey area. This figure is available in colour in the online version of the journal on Synergy.

Both Clouds have double peaked LFs with similar shapes and wavelength dependences. The exact overlap of the faint ends of the SMC and area normalized LMC distributions in JHKS shows that there is a severe sensitivity cut off for J > 14.2 mag, H > 13.5 and KS > 13.3 mag. On the other hand, the difference for I suggests the drop of the OGLE II detection efficiency occurs at fainter magnitudes, so that the faint limit in our sample is set by 2MASS and not by OGLE.

An interesting point is the changing shape of the AGB peak. It is the narrowest in I[where the full width at half maximum (FWHM) is 0.7 mag] and going toward the redder bands the stars spread over wider and wider luminosity ranges (the FWHM increases to 2 mag). This can be understood qualitatively in terms of the larger bolometric correction for the brightest and reddest stars (Alvarez et al. 2000), which results in the observed narrower distribution in I. We point out the overall similarity of the two AGB LFs, with only minor differences.

The fainter peaks, on the other hand, show quite different behaviour. The FWHM stays essentially constant with wavelength, with slightly narrower distributions for the SMC. Since the RGB luminosity function is increasing monotonically toward fainter magnitudes (Nikolaev & Weinberg 2000), the sharp cut off in our data is a consequence of the falling detection efficiency in the combined OGLE II+2MASS data base. The dip between the brighter and fainter peaks is very close to the TRGB magnitudes determined by Cioni et al. (2000) from an analysis of about 150 000 objects in the DENIS catalogue, and that close proximity was interpreted in both Ita et al. (2002) and Paper I as the evidence for pulsating RGB stars.

Following the referee's recommendations, we examined this agreement more closely. Owing to the nature of the tip of the RGB, it manifests as an edge in the luminosity function (ƒobs). Cioni et al. (2000) have shown very thoroughly in their Appendix that making an unbiased estimate of the discontinuity can be very difficult. Previous authors generally determined the position of the peak in ƒobs, while Cioni et al. (2000) argued for using ƒobs. Adopting their considerations, we determined the second derivatives of smoothed luminosity functions in all bands (Fig. 3). The OGLE II LFs were smoothed with a Gaussian weight function (FWHM = 0.04 mag) and the derivatives were approximated by the difference ratios.

Figure 3.

The second derivatives of the smoothed luminosity functions. The arrows point to the maxima for the LMC (solid lines) and the SMC (dashed lines). This figure is available in colour in the online version of the journal on Synergy.

Figure 3.

The second derivatives of the smoothed luminosity functions. The arrows point to the maxima for the LMC (solid lines) and the SMC (dashed lines). This figure is available in colour in the online version of the journal on Synergy.

Figure 4.

P–L relations in the SMC as function of the full amplitude of modes. Three JKS colour ranges were selected to plot in different colours. Light grey (turquoise in the online version): 0.9 < JKS ⩽ 1.2; black (blue in the online version): 1.2 < JKS ⩽ 1.4; dark grey (red in the online version): JKS > 1.4. This figure is available in colour in the online version of the journal on Synergy. Symbols in the lower panels are drawn larger for improved clarity.

Figure 4.

P–L relations in the SMC as function of the full amplitude of modes. Three JKS colour ranges were selected to plot in different colours. Light grey (turquoise in the online version): 0.9 < JKS ⩽ 1.2; black (blue in the online version): 1.2 < JKS ⩽ 1.4; dark grey (red in the online version): JKS > 1.4. This figure is available in colour in the online version of the journal on Synergy. Symbols in the lower panels are drawn larger for improved clarity.

The positions of maxima are listed in Table 2. For comparison, we also show the dips of the LFs and the TRGB values determined by Cioni et al. (2000). The overall agreement is excellent: most of the differences do not exceed the uncertainties. The largest discrepancy was found in KS, which can, however, be explained by a ∼0.1 mag systematic shift between 2MASS and DENIS K magnitudes. Therefore, we conclude the luminosity functions of OGLE II variables show exactly the same behaviour as, for instance, those of the much larger DENIS Catalogue towards the Magellanic Clouds, which included both variable and non variable stars.

Table 2.

Summary of TRGB magnitude values (the uncertainty is about 0.04 mag). C2000 refers to TRGB values in Cioni et al. (2000).

Table 2.

Summary of TRGB magnitude values (the uncertainty is about 0.04 mag). C2000 refers to TRGB values in Cioni et al. (2000).

Additionally, we find a significant wavelength dependence of the TRGB values, which follows exactly the changes observed in the tip of the RGB as a function of metallicity in globular clusters by Ferraro et al. (2000) and Ivanov & Borissova (2002). For instance, Ferraro et al. (2000) derived MtipK∼−0.6 [Fe/H], which implies ΔK≈ 0.18 mag, assuming the metallicity differs by a factor of 2 between the Clouds. The observed differences from the second derivatives are 0.19 mag, 0.25 mag and 0.23 mag in J, H and KS, respectively (adopting Δμ0 = 0.44 mag and correcting for differential extinctions) and they are also in good agreement with the relations in fig. 4 of Ivanov & Borissova (2002). In the I band, the difference is only 0.03 mag, which illustrates the insensitivity of MTRGB(I) to metallicity and age (Lee, Freedman & Madore 1993).

We interpret these agreements as our last piece of evidence that the dip between the AGB and RGB peaks indeed corresponds to the tip of the RGB. In Paper I, we have also considered the arguments of Alves et al. (1998) and Wood (2000) that all variables below the TRGB are thermally pulsing AGB stars and that their brighter cut off matches the TRGB by coincidence. With the presented behaviour, this explanation seems to be quite unlikely. Even if there are faint thermally pulsing AGB (TP AGB) stars below the tip, their fraction must be small compared with first ascent red giants.

Amplitude distributions

We finish the comparative analysis by presenting the same six slices of the (period, amplitude, KS magnitude) data cube as were shown for the LMC in fig. 4 of Paper I. Colours refer to three JKS colour ranges, enabling a rough classification of stars as ‘hot’, ‘warm’ and ‘cool’ red giants.

The distributions are similar to those of the LMC in that there is a good correlation between the amplitude and mode of pulsation. For amplitudes larger than 0.04 mag, practically all variables below the TRGB disappear. Furthermore, the colour distribution within any particular P–L ridge follows the same pattern as in the LMC. There are, however, some striking differences. First, there is a relative lack of ‘warm’ stars among the low amplitude variables of the SMC. There is one blue dot for every 30 dots in the upper left panel; in the LMC, the ratio is 1: 10. A similar underabundance is also visible in the lower panels. Secondly, the lower three panels are dominated by the very red stars with JKS > 1.4 mag, which are probably carbon stars. For the highest amplitude variables (all with regular Mira like light curves), there is only a handful of ‘hot’ and ‘warm’ giants, many of which fall below the fiducial Mira P–L by as much as 1.5–2 mag. These ‘faint Miras’ be identified with the dust enshrouded Mira stars (Wood 1998), since their extinctions increase strongly toward shorter wavelengths, reaching up to 4–6 magnitudes in I. Our data illustrate nicely that, despite the fact that carbon rich Mira stars obey the same P–L relation as the oxygen rich Miras (Feast et al. 1989), the use of the Mira P–L relation in extragalactic distance measurements must be restricted to the bluest possible Mira stars.

Conclusions

In this paper we presented the results of a period analysis of OGLE II red variable stars in the SMC. Multiple periodicity has been found for the majority of stars, and can now be identified as common behaviour in pulsating red giants, regardless of the metallicity.

We confirm the existence of distinct P–L relations below the tip of the red giant branch, announced in Paper I for the LMC. Utilizing the overall P–L distributions, we determined multiwavelength relative distance moduli and extinctions, which support a consistent view that the differential distance modulus of the Clouds is Δμ0≈ 0.44 mag. The luminosity functions revealed the AGB variables follow very similar distributions in both Clouds. We showed that the dips between the AGB and supposed RGB peaks do indeed correspond to the tip of the RGB (within a few hundredths of a mag). This is also supported by the wavelength dependent ΔMTRGB (∼0.20 mag in JHKS) found as an excess to the relative distance modulus. It is consistent with the metallicity dependence seen among globular clusters and we take this as further evidence that the contamination by thermally pulsing AGB stars is likely to be small.

Acknowledgments

This work has been supported by the FKFP Grant 0010/2001, OTKA Grant #F043203 and the Australian Research Council. Thanks are due to an anonymous referee, whose suggestions led to significant improvement of the paper. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The NASA ADS Abstract Service was used to access data and references.

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Author notes

On leave from University of Szeged, Hungary.