HIGH-AMPLITUDE δ-SCUTIS IN THE LARGE MAGELLANIC CLOUD

The SuperMACHO project is a five-year optical survey of the LMC aimed at detecting microlensing of LMC stars (Rest et al. 2005). The goal of this survey is to determine the location of the lens population responsible for the excess microlensing rate observed toward the LMC by the MACHO project (see Alcock et al. 2000b, and references therein) and thereby better constrain the fraction of MACHOs in the Galactic halo. The survey was conducted on the CTIO Blanco 4 m telescope using a custom VR broadband filter with the MOSAIC II wide-field imager. Garg et al. (2007), Miknaitis et al. (2007), Garg (2008a), and Rest & Garg (2008) provide a more complete description of the survey observations and data reduction. Here, we provide a brief overview of key elements of the observations and data reduction. During each year of the survey, SuperMACHO observed 68 LMC fields over ∼30 half-nights during dark and gray time between the months of September and January. The nights and the data reduction pipeline were shared with the ESSENCE survey (Miknaitis et al. 2007; Wood-Vasey et al. 2007). The SuperMACHO survey was completed in 2006 January.

Variable objects are identified using the difference imaging algorithm developed by Alard and Lupton (Alard & Lupton 1998; Alard 2000). The technique is implemented using the High-Order Transform of PSF and Template Subtraction (HOTPANTS) software package.¹⁸ Excess or negative flux, "difference flux," detections with a signal-to-noise ratio (S/N) > 5 are matched between images. Those coincident within 054 (2 pixel) of each other from image to image are assumed to belong to the same source, and the average position of these detections is taken as the source position. If there are at least three detections of a source in any survey year, we identify the source as a "real" variable object.

Once we have identified a source as variable, we perform fixed-centroid, or forced, photometry at the detection centroid to obtain a complete difference flux light curve (Garg et al. 2007; Garg 2008a). We measure the difference flux using a modified version of DoPHOT (Schechter et al. 1993) that identifies sources of negative flux. Because there are few actual sources of difference flux and many artifacts in any difference image, we cannot use the difference image itself to determine the point-spread function (PSF). Instead, we use the PSF determined for the science image prior to image differencing. We also force the centroid to be at the position determined by the high S/N difference flux detections.

Because we neglect covariance terms when convolving the images for differencing, we find that we underestimate the noise in the difference image. To empirically correct for this, we obtain the flux and its uncertainty for a grid of positions across the difference image using aperture photometry. If the flux uncertainties reflect the Poisson noise, then we would expect a histogram of flux/δ-flux to exhibit a Gaussian distribution centered at 0.0 with a standard deviation of 1.0. For a typical image, we find the standard deviation to be closer to 1.5, so we adjust our flux uncertainties so that they give a standard deviation of 1.0 which provides a more accurate estimate of the uncertainty in the measured flux (Garg 2008a). We also find that because it uses an analytical PSF model that does not reflect additional structure in the PSF, DoPHOT further underestimates the uncertainty in the measured flux by 0.01 mag (Garg 2008a). We add this term in quadrature to the adjusted uncertainty returned by our modified DoPHOT to obtain the uncertainties reported in this paper.

In addition to the VR survey images, we also obtained a set of high-quality B- and I-band images. We process these images using the reduction pipeline described above. We create B and I catalogs for these images and generate B − I color–magnitude diagrams (CMDs) for each field. Though the MOSAIC II camera does not allow for simultaneous imaging in multiple bands, these observations were typically made within 5 minutes of each other. The colors for even relatively short-period variables (1–2 hr) should be sufficiently accurate to determine the rough position of sources in the CMD.

After identifying all variable sources detected by SuperMACHO, we phase their light curves to find periodic variables. We perform light-curve phasing using both the SuperSmoother algorithm (Reimann 1994) and the CLEANest code (Foster 1995). For SuperSmoother, we use the period that gives the SuperSmoother curve with the smallest residuals. For CLEANest, we use the period corresponding to the highest power frequency from the light curve's frequency spectrum. For the initial candidate selection, we take the shorter of the two periods to be the light curve's period, P. For the objects presented in this paper, this is typically the period determined by CLEANest. For all periodic variables, we find the mean of the difference flux light curve (which may be negative). We add this to the template flux to determine the mean magnitude of the variable object, VR.

We present 2323 light curves of HADS in the LMC discovered by the SuperMACHO project. We use the following criteria to select these objects.

• 1.  
  Using the values of P and VR from the initial phasing, we create a PL diagram. We find an overdensity of sources that lie roughly in the region expected for δ-Scutis in the LMC. Based on the PL diagram, we define this region as 19.7 < VR < 22.2 and 0.045 < P < 0.145.
• 2.  
  We select a higher-amplitude subset of these candidates. We choose only light curves for which the difference between the brightest and faintest data points is greater than 0.2 mag.
• 3.  
  We cross-match the set of light curves against the B and I catalogs described in Section 2.4 using a match radius of 027 (1 pixel). We remove candidates that do not have matches in both bands. We also remove candidates that do not lie in the main-sequence portion of the CMD. To ensure this, we require each candidate to have I > 19.5 mag and 0.0 mag <B − I < 1.3 mag. Figure 1 shows the CMD for all SuperMACHO sources falling on a single amplifier and the location of the δ-Scuti candidates. After applying these selection criteria, we have 4126 candidate sources.
• 4.  
  We use the SigSpec (Reegen 2007) code to determine the frequency spectrum of the light curve (see Section 4.2). In many cases, the primary frequency of variation found by SigSpec differs from that found during the initial phasing. We select only light curves having a primary frequency of variation with spectral significance, f_sig, greater than 5.475, which roughly corresponds to an S/N of 4 (see Reegen 2007). For the remainder of the analysis, we use the frequency associated with the highest amplitude of variability found by SigSpec such that P = 1/f_SigSpec. Using the newly determined periods, we select only candidates with 0.045 days <P < 0.115 days.¹⁹
• 5.  
  We perform a fourth-order Fourier decomposition of the light curves (see Section 3.4). We use the coefficient of the zeroth-order term, a₀, as the average stellar magnitude and keep candidates with 19.8 < a₀ < 22.2.
• 6.  
  We determine an overall amplitude using the model curve described by the fourth-order Fourier fit. We define the overall amplitude, Δ, as the difference between the minimum and maximum points in the model curve, or the peak-to-trough amplitude. We select only those candidates with a Δ greater than 0.2 mag. This yields a final set of 2323 HADS candidates. The majority of the candidates eliminated after applying the CMD criteria are eliminated by this amplitude cut.

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To determine the efficacy of the above selection criteria, we have inspected the light curves by eye. We estimate that <2% are contaminants or poorly phased. To retain objective selection criteria, however, we leave them in the final set. Table 1 gives the position, light-curve characteristics, and color information for all the HADS candidates. Table 2 gives the light curves for all candidates. Figure 2 shows a selection of candidate light curves.

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We compare our set of δ-Scutis to sets obtained from the OGLE-II (Soszyński et al. 2003) and MACHO surveys (Alcock et al. 2000b). McNamara et al. (2007) analyze a catalog of non-RR Lyrae pulsating variables accompanying the main Soszyński et al. (2003) catalog to find evidence of 24 δ-Scuti candidates in the LMC. We cross-match the McNamara et al. (2007) catalog against the set of all SuperMACHO variables, including those we do not classify as δ-Scutis. We find only one match within 20 of the position reported in McNamara et al. (2007). This object, with OGLE-II designation 050309.65-684327.6, also appears in our candidate set as 7151_smc8_15 and lies 154 from the OGLE-II position. Lepischak (2007) reports 101 HADS discovered in a subset of the MACHO database. Of these, 99 fall within the field of view (FOV) observed by SuperMACHO. We note that though these objects fall within the SuperMACHO FOV, several may lie within gaps between amplifiers or close to saturated sources that have been masked. We estimate that these considerations eliminate roughly 5% of our FOV. We cross-match the Lepischak (2007) catalog to the set of SuperMACHO variables and find 37 matches within 2'' of the MACHO positions. Of these, 17 appear in our set of δ-Scuti candidates. A majority of the eliminated matches are removed based on the third criterion described above which requires candidates to lie on the main sequence described by the B − I CMD. We find, however, that without this criterion many of the candidates lie along the red branch and have light curves similar to close eclipsing binaries.

We consider why we find so few matches to these catalogs. Some of the sources identified in these catalogs may not meet the variable object selection criteria described in Section 2.2. We find, however, that a cross-match to the set of Blazhko RR Lyrae identified in MACHO (Alcock et al. 2000c) yields matches within 2'' for 62% of the MACHO sources in the SuperMACHO FOV. The larger match fraction to the MACHO RR Lyrae catalog suggests that low detection efficiency is unlikely to account for the entire discrepancy.

We also examine some of the template images at positions where δ-Scutis detected in MACHO were not found in SuperMACHO. We do not see any sources in the templates within several arcseconds of these positions. This suggests that the discrepancy does not result from a failure to classify these objects as variable but rather from a lack of sources at these positions. This may suggest differences between our astrometric solutions, though we emphasize that we do find at least one match within 2'' in each catalog. If we allow for a much larger match radius of 5'', we find 44 matches to Lepischak (2007) and 3 to McNamara et al. (2007). Of these matches there are no new candidates that pass our selection criteria. We note that the apparent brightnesses of LMC δ-Scutis are very close to the detection limit for both MACHO and OGLE-II. Misclassification of these faint sources would not be surprising in these catalogs.

In addition to the LMC HADS reported by MACHO and OGLE-II, HADS have been reported by the OGLE-III collaboration (Soszyński et al. 2008; Soszyński 2009). Notably, the OGLE-III PL diagram suggests that the HADS have longer periods than those discussed in this work. This will be an important area for further investigation when the final catalog becomes available.

Using the period determined by SigSpec, we perform a fourth-order Fourier decomposition of the light curve to obtain the coefficients in the series

$$\begin{equation} {V\!R}(t) = a_{0} + \sum _{n=1}^{4} a_{n} \cos \left(\frac{2 \pi n t}{P}\right) + b_{n} \sin \left(\frac{2 \pi n t}{P}\right), \end{equation} \tag{ 1 }$$

where VR(t) is the observed VR magnitude at time t, n counts each order in the decomposition, a_n and b_n are the coefficients of each order determined by the fit, and P is the period. We find that a fourth-order expansion is sufficient to capture the structure of the light curves.

Using the Fourier coefficients, we calculate several additional parameters describing the light curves. A_n is the amplitude of pulsation for each order such that A²_n = a²_n + b²_n. r_ij is the ratio of amplitudes such that r_ij = A_i/A_j. ϕ_n is the phase shift for each order such that ϕ_n = atan(b_n/a_n). ϕ_ij is given by ϕ_ij = ϕ_i − iϕ_j. These parameters are given in Table 3 for all candidates.

Figure 3 shows the PL diagram of the final HADS candidates. We note that the PL relation shows a high degree of scatter. This may indicate subgroups within the sample such as the subluminous group observed by P08. To identify such groups, we create a histogram of the PL-relation-corrected luminosity, VR_c, of the candidates similar to P08 (Figure 4). We assume the slope of the V-band PL relation reported by P08 without any transformation or metallicity correction. This gives the relation:

$$\begin{equation} {V\!R}_{c,-3.65} = 3.65 \log _{10}P + {V\!R}. \end{equation} \tag{ 2 }$$

Using this equation, we fit for the PL relation for the SuperMACHO δ-Scutis. Because of the possibility of subgroups, we perform the fit using the "ridgeline" of the PL diagram. This assumes that the main population of δ-Scutis is the most populous. To find the "ridgeline," we first bin the data by period (0.05 day binsize) and then by VR (20 bins for each period bin). We then take the densest VR bin for each period bin as the mode. Figure 5 shows the median and mode magnitudes for each bin. Using the mode as the dependent variable and the inverse square root of the number of candidates in each period bin as the uncertainty, we perform a linear, least-squares fit to the data. This yields the PL relation:

$$\begin{equation} {V\!R}= -3.65 \log _{10}P + 16.68\pm 0.11\, {\rm mag}. \end{equation} \tag{ 3 }$$

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We also independently determine the slope of the PL relation using the SuperMACHO δ-Scutis. We again perform a fit to the ridgeline, this time leaving the slope as a variable parameter. This fit yields the relation:

$$\begin{equation} {V\!R}= -3.43\pm 0.26 \log _{10}P + 16.98\pm 0.30. \end{equation} \tag{ 4 }$$

Similar to Equation (2), we determine the intercept, VR_c,−3.43, for all candidates using the slope from the best fit. Figure 4 shows the histogram of VR_c,−3.43 (hereafter, VR_c refers to the PL-corrected-luminosity using this best-fit slope). We examine the impact of our binsize on the best fit to the ridgeline. We find that both doubling and halving the binsize give shallower slopes. In both instances, however, this is because the selected binsize gives too much weight to the sparse region of the PL diagram at short periods.

In addition to the primary frequency of variation, SigSpec finds lower amplitude modes of variation. We use the SigSpec frequency spectrum to determine whether the candidates show multiple modes of pulsation. We consider any frequency within $\frac{1}{10}$ of a whole number ratio to the primary frequency to be an alias. We only consider secondary frequencies with a spectral significance greater than 5 (see Reegen 2007). SigSpec does return additional modes with significance greater than 5 for a few candidates. Given the typical S/N and sampling of the SuperMACHO data, however, we do not consider these to be reliable.²⁰ Figure 6 shows the ratio between the primary and secondary frequency, f_primary/f_secondary, as a function of the spectral significance of the secondary. Table 3 gives f_primary/f_secondary for candidates showing a secondary frequency.

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Examination of Figure 6 reveals two distinct groups with high values of f_sig. We find 119 candidates in the first group with 0.765 < f_primary/f_secondary < 0.790, and we find 19 candidates in the second group with 1.275 < f_primary/f_secondary < 1.305 (Figure 7 shows a representative selection of light curves for both groups). The frequency ratios indicate that these groups are multimode fundamental (F) and first overtone (FO) pulsators, respectively (see McNamara et al. 2007; Poretti et al. 2005; and references therein). We note that there are several other light curves with f_{sig,secondary} greater than 5.0. Inspection of their light curves reveals that while they do show additional scatter, it is not as pronounced as that of the candidates falling into the groups described above (Figure 8 shows a representative selection of these light curves). This may contribute to greater uncertainty in determining the secondary frequency.

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Figure 9 shows the Petersen diagram for the multimode F and FO candidates. We have assumed that the candidates with frequency ratios greater than one are varying in overtone modes, and set P₀ as their secondary period. We note that we observe several candidates with ratios above 0.778 whereas Poretti et al. (2005) observe only one. Based on their findings, this is likely an indication that these candidates are metal-poor compared to the Galactic sample. Such an interpretation is consistent with observational measurements of LMC metallicity (Cole et al. 2005).

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We note that the multimode pulsators are concentrated toward fainter values of VR_c. We examine whether the lack of multimode pulsators at the brightest values of VR_c may be attributable to lower sensitivity to secondary periods for these candidates. A plot of Δ against VR_c (Figure 10) reveals that the brighter sources generally have smaller amplitudes. To determine our ability to detect secondary modes of pulsation, we simulate several double-mode sinusoidal light curves as they would appear in our data (see Garg 2008a, 2008b and A. Garg et al. 2010, in preparation). We model the light curves using

$$\begin{equation} {V\!R}(t) = A_0 + A_1 \sin (t f_1) + A_1\, A_{\rm ratio} \sin (t f_{\rm ratio}\, f_1), \end{equation} \tag{ 5 }$$

where VR(t) is the VR magnitude at time t, A₀ is the mean magnitude, A₁ is the amplitude of the primary frequency, A_ratio is the ratio between the amplitude for the primary and secondary frequencies, f₁ is the primary frequency, and f_ratio is the ratio between the primary and secondary frequency. All parameters except f_ratio are randomly selected from a uniform distribution. We choose A₀ to be between 19 and 22.5. We choose A₁ between 0.07 and 1.1. We choose A_ratio between 0.1 and 1.0. We restrict f_ratio to be either 0.777 or 1.288, corresponding to the ratios for multimode F and FO pulsators, respectively. We note that using our definition, Δ is equal to twice A₁ in these simulations. We find that our ability to detect secondary modes does not depend strongly on A_ratio. For values of A_ratio close to 1.0, we do occasionally misidentify the secondary period as the primary, but overall our detection efficiency is roughly 65%–80% for all values of A_ratio integrated over Δ. Using the values of A₀ and f₁, we determine VR_c for each simulated light curve by Equation (4). We also find no strong correlation between VR_c and our efficiency for detecting secondary modes, though the fainter light curves tend toward the low end of the efficiency range. We do find that our efficiency for detecting secondary modes depends on the amplitude of the primary mode of pulsation, A₁. For A₁ close to 0.1, corresponding to Δ close to 0.2, our detection efficiency for multimodes in FO pulsators ranges from 15% to 50%. For multimode F pulsators, it is higher and ranges from 40% to 60%. For no other parameter do the detection efficiencies differ so greatly between F and FO frequency ratios.

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The histograms shown in Figure 4 are heavily skewed toward brighter magnitudes. Typically, such a population that lies above the main PL relation is thought to be pulsating at the overtone frequency (see P08 and references therein). We also find that the multimode FO pulsators identified in Section 4.2 are brighter than the multimode F pulsators, though there is no evidence of secondary pulsation modes in the brightest candidates to help confirm the FO interpretation. As discussed above, we have a low efficiency for detecting multimode overtone pulsators at small amplitudes. We note that Figure 10 indicates that the brightest values of VR_c correspond to the lowest amplitude candidates. Based on observations and models of RR Lyrae, small amplitudes are consistent with the overtone hypothesis (Bono et al. 1996). As discussed in Section 4.2, small amplitudes also reduce our ability to detect multiple modes particularly in FO pulsators. While the presence of multimode FO pulsators among the brightest candidates would lend additional support to the overtone hypothesis, their lack does not rule out this interpretation.

We also examine the shape of the phased light curves using the Fourier components. Based on observations of FO RR Lyrae, we would expect overtone pulsators to exhibit a more symmetric light curve (Stellingwerf et al. 1987; McNamara 2000). By examining the relation between r₂₁ and VR_c (Figure 11), we find this to be the case. Because ϕ₂₁ is relatively constant for all candidates (Figure 12), r₂₁ measures the relative contributions of the second and first Fourier components to the overall amplitude. More asymmetric light curves should have higher values of r₂₁, and we find that lower values of VR_c correspond to lower values of r₂₁. This is suggestive that the brighter sources with lower values of r₂₁ may be overtone pulsators having more symmetric light curves.

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We plot the difference between the phase of minimum and maximum, ϕ_diff, against VR_c (Figure 13). For a purely symmetric, sinusoidal light curve, we would expect ϕ_diff to be 0.5. We find that the majority of our candidates lie between 0.35 and 0.4. Notably, however, the FO multimode candidates have larger values of ϕ_diff corresponding to a more symmetric light curve as we would expect. We find, however, no strong correlation between ϕ_diff and VR_c.

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By examining the amplitude and shape of the brightest candidates and comparing to trends observed in RR Lyrae, these light-curve analyses indicate that we may have several candidates pulsating in the FO mode. Notably, Bono et al. (1997) conclude that the effect may be reversed for δ-Scutis: overtone δ-Scutis may exhibit larger amplitudes and more asymmetric light curves. Our data disagree with this finding, however, as the small amplitude candidates examined in this section lie above the main PL relation, indicating that they have shorter periods than the majority of the candidates with similar luminosities.

We also note that unlike FO RR Lyrae and FO Cepheids, we see no clear separation between the F and possible FO δ-Scutis. Our results are similar to those of P08 whose PL diagram also does not show a clear separation. We consider whether the lack of separation indicates significant scatter in the metallicities of the candidates. Such scatter would shift the intercept of the PL relation and potentially smear the separation between fundamental and overtone pulsators. Cole et al. (2005) do find a significant spread in LMC metallicity based on spectroscopic observations of the Calcium II triplet in red giants toward the bar region. They find a primary population with [Fe/H] = −0.37 and σ = 0.15 and a second, metal-poor population with [Fe/H] = −1.08 and σ = 0.46. Using the metallicity dependence of 0.19 [Fe/H] from McNamara et al. (2004), we might expect an additional 0.03 mag of scatter in the PL relation based on these results.

Differing extinction may also contribute additional scatter. If we assume the redding distribution is E(B − V) = 0.13 with σ = 0.045 (Harris et al. 1997) and R_v = 3.1, we expect an additional 0.14 mag of scatter from variations in extinction. The tilt of the LMC will also contribute additional scatter of ∼0.01 mag (Sebo et al. 2002). We examine the uncertainty in our period determinations as this may also contribute to additional scatter. Using the technique used for the multimode light-curve simulations described in Section 4.2, we simulate δ-Scuti light curves as they appear in our data. We use the Fourier coefficients from our candidate δ-Scutis and Equation (1) to model the light curves. We find that the periods we measure for more than 80% of the simulations we recover are within 0.001 days of the input period. The additional scatter contributed from this uncertainty is negligible.

We add in quadrature the additional scatter contributed to VR_c by the above factors to obtain an expected scatter of 0.14 mag. Inspection of Figure 4, however, reveals that the difference between the mode of VR_c,−3.43, 17.0 mag, and the next-densest bright bin, 16.7 mag, is 0.3 mag. This difference is more than 2 times the additional scatter, large enough that it is difficult to explain the lack of a clear separation between F and FO pulsators based on these factors alone. These data may indicate that a clear separation between the PL relation for F and FO modes does not exist for δ-Scutis. This is not necessarily surprising. δ-Scutis lie in a region of the instability strip that spans a wide range of temperatures and luminosities. Because of this, we would expect the PL diagram to have a broad intrinsic width resulting in regions of FO pulsation that overlap regions of F pulsation.

The histogram of VR_c shows no strong indication of the subluminous population described in P08. We find, however, that the median Δ increases for fainter sources (Figure 10). To test whether this reflects a selection bias against smaller amplitudes for fainter sources, we use the simulations of δ-Scuti light curves described in Section 4.3. We find that while we have a relatively high efficiency for detecting δ-Scutis at all amplitudes and luminosities considered (>65%), for the faintest sources we are ∼13% more efficient at detecting larger amplitude (0.8 mag <Δ< 1.0 mag) variables than smaller amplitude (0.2 mag <Δ< 0.4 mag) variables. The ratio between detection efficiencies for different amplitudes is similar for brighter sources. Figure 14 shows the median and 33rd percentile error bars after correcting for detection efficiency. We find that the median amplitude still increases with VR_c.

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We also find that the majority of sources with Δ> 0.4 mag have VR_c fainter than 16.8 mag. We note that the sources in P08 also have larger amplitudes, generally greater than 0.4 mag. A histogram of only the SuperMACHO LMC candidates with Δ> 0.4 mag also indicates an excess of subluminous sources similar to that observed by P08 (Figure 15). We suggest that the subluminous population described in P08 may reflect these larger amplitude sources.

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As discussed in P08, the explanation for these sources remains an open question. Observations and models of RR Lyrae show that for F pulsators amplitude increases at lower metallicities (Bono et al. 1996). This may suggest that these candidates represent a more metal-poor population. We consider whether they may belong to the metal-poor LMC population observed by Cole et al. (2005) having [Fe/H] = −1.08 ± 0.46. Again using the metallicity dependence from McNamara et al. (2004) of 0.19 [Fe/H], we would expect δ-Scutis from this population to lie 0.21 mag below the main PL relation. Our data show a larger separation of almost 0.4 mag. Notably, the separation between the main and subluminous populations observed by P08 is similar. Given the differences in the compositions and star formation histories of Fornax and LMC, this similarity lends support to the conclusion that these large amplitude, subluminous sources evolved from the same population as those on the main PL relation. It would otherwise be difficult to explain why both galaxies have such similar larger amplitude, subluminous populations that evolved independently.