RR LYRAE VARIABLES IN TWO FIELDS IN THE SPHEROID OF M31 ^*

We chose to work on the FLT images as retrieved from the HST archive. These frames have been bias subtracted and flat fielded, but, unlike the drizzled (DRZ) images, they retain the geometric distortions of ACS (Mahmud & Anderson 2008). Photometry was performed using the same procedure as Sarajedini et al. (2006). The first step is the application of geometric correction images to the Wide Field Channel 1 and 2 (WFC1 and WFC2, respectively) portions of the FLT images. After this step, the data quality maps are applied where the values of the bad pixels in the science images are set to a number well below the sky background to be sure the photometry software ignores those pixels. At this point, the resultant images are ready to be photometered.

The detection of the stellar profiles and the measurement of magnitudes were done with the DAOPHOT/ALLSTAR/ALLFRAME crowded-field photometry software (Stetson 1987, 1994). After the application of the standard FIND and PHOT routines to detect stars and perform aperture photometry on them, ALLSTAR was applied to each of the 32 images in order to derive well-determined positions for all of the stars. In this step and in subsequent ones involving the application of a point-spread function (PSF) in order to determine positions and magnitudes, we made use of the high signal-to-noise PSFs constructed by Sarajedini et al. (2006). The reader is referred to that paper for the details of the PSF construction process.

The stellar positions from the ALLSTAR runs were used to construct a coordinate transformation between each of the 32 images and these were used to combine all of the images into one master frame per field. This combined frame was then input into ALLSTAR, from which a master coordinate list of stellar profiles was constructed. The resultant coordinate list along with the spatial transformation between the images and the PSFs were used in ALLFRAME to derive magnitudes for all detected profiles on each image. At this point, the measurements on each of the individual frames were matched and only stars appearing on all 32 images were kept.

The standardization of the individual magnitudes proceeded in the following manner. First, the correction for the charge transfer efficiency problem was applied using the prescription of Reiss & Mack (2004). The magnitudes were then adjusted to a radius of 0.5 arcsec and corrected for exposure time. Offsets to an infinite radius aperture published by Sirianni et al. (2005) were then applied. Finally, the resultant values were calibrated to the VegaMAG system using the zero point for the F606W filter from Sirianni et al. (2005). A correction to this zero point amounting to 0.022 mag was applied as a result of a revised calibration of the ACS/WFC photometric performance by Mack et al. (2007). Each of our magnitude measurements is affected by three sources of systematic error: the uncertainty in the aperture corrections (±0.02 mag), the error in the correction to infinite aperture (±0.00 mag) for the F606W filter, and the error in the VegaMAG zero point (±0.02 mag).

For a given star with 32 magnitude measurements, we calculated the mean photometric error as returned by ALLFRAME (〈σ_err〉) and the standard deviation of the measurements (σ_sd). For the first round of variable searching, we considered any star a candidate if σ_sd / 〈σ_err〉⩾3.0. Approximately 3000 stars fit this criterion in each of our two fields.

These stars were then input into our template-fitting period-finding algorithm, which is based on the method used by Sarajedini et al. (2006) as originally formulated by Layden & Sarajedini (2000). We have taken the FORTRAN code written for the Layden & Sarajedini (2000) study and rewritten it using the Interactive Data Language (IDL) incorporating a graphical user interface (GUI). The original FORTRAN code used the "amoeba" minimum-finding algorithm exclusively, but our code, dubbed FITLC,⁶ has the option to use a more robust algorithm known as "pikaia" which has its roots in the study of genetics. The software uses 10 template light curves—six ab-type RR Lyraes, two c-type RR Lyraes, one eclipsing binary, and one contact binary. It searches over a period range from 0.2 day to a specified maximum (2.2 days for our Field 1 and 3.1 days for Field 2) looking for the period that minimizes the value of χ². This is accomplished with a two-step process. First pikaia is used to find the combination of epoch, amplitude, and mean magnitude that minimize χ² at evenly spaced period increments of 0.01 day. Then pikaia is applied again to find the combination of epoch, amplitude, mean magnitude, and period that minimize χ² within ±0.01 day of the period with the lowest χ². The best fitting period from this final step is taken to be the period of the variable. The resultant phased light curves for each star were visually examined, and the stars that presented a compelling case for variability were retained in our final database. Of the ∼6000 total stars originally fit, 752 exhibit genuine variability as shown by our data.

As a test of our template-fitting method, we have also applied the Lomb-Scargle period-finding algorithm (Scargle 1982; Horne & Baliunas 1986) to the time series photometry of the RR Lyraes in our sample. We find a mean difference of 0.0007 d in the periods determined by the two methods throughout the period range of RR Lyraes. In the minority of cases where template-fitting and Lomb-Scargle yield significantly different results, the resultant phased light curves are of significantly higher quality for the former method as compared with the latter. Furthermore, to test for the presence of aliasing effects in our derived periods, we have also examined fitted light curves using periods that correspond to χ² minima near half of the optimum period. In all cases, these fits are clearly inferior to the ones yielded by the optimum period from FITLC.

Tables 2 and 3 list the individual F606W magnitudes of each variable at each epoch wherein 2,450,000 has been subtracted from the epoch value, while Tables 4 and 5 list the candidate variables in our two fields along with their properties such as period, amplitude, and mean intensity-weighted magnitude. These stars fall into two broad categories. First, there are those that clearly show variability, but their periods are comparable to or longer than our observing window. These are referred to as "long period" in Tables 4 and 5. There is also one candidate anomalous Cepheid in our data set, which is so indicated in Table 5. The second category includes stars that exhibit clear periodic variability with a period that is significantly shorter than our observing window but longer than 0.2 d. These are the stars for which we can be confident of our periods. Some examples of stars with periods longer than our observing window are shown in Figure 2, while Figure 3 shows phased light curves of a number of contact and eclipsing binaries in our data set. Figure 4 displays the phased light curves of all of the RR Lyrae variables for which we have derived periods.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

View figure set (35 panels)

All of the stars that exhibit RR Lyrae light curves also have apparent magnitudes in the range one would expect if they are at the distance of M31. Therefore, it is reasonable to assume that most if not all of these objects are RR Lyrae stars belonging to M31 and/or its environs. This assertion will become clearer when we compare the luminosity function (LF) of the nonvariable stars with that of the RR Lyraes.

It should be noted that we are much less confident about the properties of the eclipsing and contact binaries that we have identified as compared with the RR Lyraes. This is because we know what period range to expect for the RR Lyraes (∼0.25 d to ∼0.90 d), so that our observing window provides coverage of multiple cycles of variation for a given RR Lyrae star. In contrast, the periods of the eclipsing and contact binaries cannot be similarly constrained, so it is difficult for us to ensure that our observing window is sufficient to derive the periods of these variables. As such, we have provided information for these stars (positions, periods, amplitudes, and magnitudes) so that future observers can confirm the nature of their variability, but we will not consider them further here. Instead, for the remainder of this paper, we will limit our discussion to the 681 RR Lyrae variables in our sample and what they reveal about the properties of the M31 system.

In order to characterize the possible biases in the derived periods of our variable star sample, we have performed simulations of our light-curve-fitting technique in the following manner. For the ab-type RR Lyraes, we selected one of the light curve templates (the results are insensitive to the actual ab-type template used) and produced artificial variables with a period range of 0.45–0.80 days and amplitudes between 0.3 and 1.3 mag. For the c-type RR Lyraes, a period range of 0.25–0.40 days and an amplitude range of 0.2–0.5 mag were used. One thousand variables were generated in each case, and the mean photometric error at the level of the RR Lyraes was used to populate the light curves using the same observing window as the actual data. These simulated light curves were input into our template light-curve-fitting software. We are interested in comparing the input periods with the output periods in order to gauge any possible biases present in our analysis method.

Figures 5 (RRab) and 6 (RRc) show the result of these fits for Field 1 wherein the upper panel shows the mean difference between the input and output periods, while the lower panel illustrates the differences in the period distributions. The results of the simulations for RR Lyraes in Field 2 are indistinguishable from those in Field 1. Of the 1000 ab-type RR Lyraes generated, none were mistaken for any other type of variable among the 10 light curve templates used in the fitting. For the c-type light curves, two of the 1000 generated variables were fit with a contact binary template. We find no significant biases in our determination of the periods for both types of RR Lyraes. As such, we will not apply any sort of correction to our derived periods. As for the errors in the period and amplitude determinations, these simulations suggest that an individual ab-type RR Lyrae has an error of ±0.005 day and ±0.044 mag, respectively. For a c-type RR Lyrae star, these error values are ±0.001 day and ±0.022 mag.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 7 displays a comparison of the LFs of the nonvariable stars in the two fields. Both distributions feature a quick rise from brighter magnitudes to fainter ones with a pronounced peak at m_F606W ∼ 25.3 representing the core-helium burning horizontal branch (HB) stars. A sudden drop in both LFs at m_F606W ∼ 27.7 suggests the onset of significant incompleteness as the limit of the photometry is approached. The fact that the HB stars are more than 2 magnitudes brighter than this completeness threshold indicates that our sample of RR Lyrae variable candidates should not be adversely biased by photometric incompleteness.

Zoom In Zoom Out Reset image size

The two panels of Figure 8 compare the LF of the nonvariable stars with those of the RR Lyraes in the two observed fields. We have used the intensity-weighted magnitudes listed in Tables 4 and 5 to construct these distributions. Gaussian fits to the regions around the RR Lyrae LF peaks yield 〈m_F606W〉 = 25.20 ± 0.04 for Field 1 and 〈m_F606W〉 = 25.22 ± 0.04 for Field 2, where the errors represent the standard error of the mean combined with the uncertainty in the photometric zero point added in quadrature. For the nonvariable stars, these peaks correspond to 〈m_F606W〉 = 25.25 ± 0.04 for Field 1 and 〈m_F606W〉 = 25.23 ± 0.04 for Field 2. The 1-σ width of these distributions is ∼0.11 mag. These numbers suggest no significant difference in the mean magnitudes of the RR Lyraes and the nonvariable stars. This serves to confirm our assertion that most if not all of the variable stars in this magnitude range are RR Lyrae variables. In addition, when we combine the RR Lyrae variables from both fields, we find a mean magnitude of m_F606W = 25.21 ± 0.01 on the VEGAmag system. The quoted uncertainty represents the standard error of the mean. When converted to the V band using the mean offset for RR Lyraes in the middle of the instability strip from B2004 of V − m_F606W = 0.08 ± 0.04, we derive 〈V(RR)〉 = 25.29 ± 0.01 (random) ±0.05 (systematic). This compares favorably with the value of 〈V(RR)〉 = 25.30 ± 0.01 based on the average for the ab-type and c-type RR Lyraes from the B2004 study (see Figure 8). We note in passing that a Gaussian fit to the LF of the B2004 RR Lyraes yields a 1σ width of 0.12 mag, which is comparable to the value for the RR Lyraes in the present study.

Zoom In Zoom Out Reset image size

Of the 681 total RR Lyrae stars in our sample, 555 (267 in Field 1 and 288 in Field 2) are of the ab-type with the remainder being c-type (57 in Field 1 and 69 in Field 2). This leads to a ratio of N_c/N_abc = 0.19 ± 0.02, which is quite different than the value of N_c/N_abc = 0.46 ± 0.11 inferred from the B2004 data. Therefore, our samples of ab-type and c-type RR Lyraes are a factor of ∼2 greater and a factor of ∼2 less, respectively, than what we would expect based on the B2004 field. Taken at face value, this suggests that the old population in the environs of M31 exhibits different pulsation properties at 4–6 kpc as compared with 11 kpc.

To evaluate the validity of this assertion, we need to address the question of incompleteness in our sample of RR Lyraes. Are there significant numbers of RR Lyraes in our fields that we have failed to identify? We begin by noting that B2004 claim that their samples of c-type and ab-type RR Lyraes are complete and not significantly contaminated by dwarf Cepheids. Therefore, we can gain some insight by comparing the amplitude distributions of the B2004 RR Lyrae variables with our sample as shown in Figure 9. This comparison suggests that the amplitude distribution of the B2004 RR Lyraes is consistent with those of our sample. Application of the Kolmogorov–Smirnov test to these distributions confirms this suggestion. The fact that, at the low-amplitude end, the two distributions are not substantially different argues that significant numbers of low-amplitude RR Lyraes are not missing from our sample.

Zoom In Zoom Out Reset image size

Another possibility of explaining the differences in the N_c/N_abc ratio between our fields and the one at 11 kpc is that our Field 1 data are significantly influenced by the stellar populations of M32 and not M31. In fact, we find N_c/N_abc = 0.18 ± 0.0.025 in Field 1 and N_c/N_abc = 0.19 ± 0.025 in Field 2, which are statistically indistinguishable from each other. In addition, the density of RR Lyrae variables and their period distributions are indistinguishable between Fields 1 and 2. For example, for the ab-type variables, Field 1 exhibits a mean period of 〈P_ab〉 = 0.553 ± 0.004, while Field 2 shows 〈P_ab〉 = 0.561 ± 0.005. In the case of the c-type RR Lyraes, the analogous values are 〈P_c〉 = 0.326 ± 0.005 and 〈P_c〉 = 0.327 ± 0.004, respectively. These values suggest that both of our fields sample the regions around M31 and are minimally contaminated by M32 RR Lyraes.

If this difference in the N_c/N_abc ratio between a radial distance of 11 kpc and 4–6 kpc in M31 is real, then it suggests that the RR Lyraes at 11 kpc are more akin to their brethren in Oosterhoff type II globular clusters, while those at 4–6 kpc are more like RR Lyraes in Oosterhoff type I clusters. This is based on the fundings of Castellani et al. (2003) who examined the N_c/N_abc ratio in Galactic globulars. They found that among the 12 clusters with 40 or more variables, 〈N_c/N_abc〉 = 0.37 for Oosterhoff II clusters and 〈N_c/N_abc〉 = 0.17 for those of Oosterhoff type I. These compare favorably with the values of 〈N_c/N_abc〉 = 0.46 for the 11 kpc field and 〈N_c/N_abc〉 = 0.19 for the 4–6 kpc field.

The Bailey diagram for our sample of RR Lyraes is shown in Figure 10 where we plot the variables in the two fields using different colors. However, it is clear that they occupy the same regions of this diagram. The ab-type RR Lyrae variables are shown with open circles, while the c-type stars are plotted as open triangles. The dashed line in Figure 10 represents the relation exhibited by the RRab stars in the B2004 field. The solid lines are the relations for Oosterhoff I and II globular clusters from Clement (2000). These lines have been adjusted to account for the difference between an amplitude in the V band and one in the F606W band. Interestingly, the B2004 RR Lyrae relation is closer to the OoI line, even though the N_c/N_abc ratio in the B2004 field is closer to that of OoII clusters. It is unclear why this should be the case.

Zoom In Zoom Out Reset image size

The B2004 line appears to be offset compared with our data suggesting slightly shorter periods for the RRab variables in our fields. This behavior is further exemplified in Figure 11 which shows the period distributions in the two fields compared with the mean periods of the ab- and c-type RR Lyraes from B2004. We find mean periods of 〈P_ab〉 = 0.557 ± 0.003 and 〈P_c〉 = 0.327 ± 0.003. For the B2004 field, these values are 〈P_ab〉 = 0.594 ± 0.015 and 〈P_c〉 = 0.316 ± 0.007. The mean periods of the c-type variables are statistically indistinguishable from each other, but the ab-types in our fields exhibit a somewhat shorter period as compared with those in the B2004 fields.

Zoom In Zoom Out Reset image size

It is well known that as the metallicity of ab-type RR Lyraes increases, their periods decrease (e.g., Sandage 1993; Layden 1995; Sarajedini et al. 2006). Thus, the period distribution of these stars (Figure 11) can be converted to a metallicity distribution using equations derived by previous investigators. Using the data of A. C. Layden (2005, private communication) for 132 Galactic RR Lyraes in the solar neighborhood, Sarajedini et al. (2006) established a relation between period and metal abundance of the form

$$\begin{eqnarray} &&[{\rm Fe/H}] = -3.43 - 7.82 {\rm Log} P_{\rm ab}. \end{eqnarray} \tag{ 1 }$$

This equation does not take into account the amplitudes of the RR Lyraes even though, as Figure 10 shows, there is a relation between amplitude and period for the ab-types. The work of Alcock et al. (2000) yielded a period–amplitude–metallicity relation of the form

$$\begin{eqnarray} &&[{\rm Fe/H}] = -8.85[{\rm Log} P_{\rm ab} + 0.15A(V)] -2.60, \end{eqnarray} \tag{ 2 }$$

where A(V) represents the amplitude in the V band. We applied an 8% increase to the m_F606W amplitudes to convert them to V band values (B2004). Figure 12 shows the metallicity distribution function (MDF) for the ab-type RR Lyraes. The top panel compares the results obtained using Equations (1) and (2), while the lower panel compares the MDFs for the two fields using Equation (2).

Zoom In Zoom Out Reset image size

We see in the upper panel of Figure 12 that, while the two MDFs exhibit very similar peak metallicities, the MDF generated using Equation (2) displays a more prominent peak. This is the representative of the fact that Equation (2) accounts for the variation of period with amplitude as well as metallicity resulting in a cleaner abundance signature in the MDF. In the lower panel of Figure 12, we see that the peaks of the RR Lyrae MDFs in our two fields differ by ∼0.1 dex; we find 〈[Fe/H]〉 = −1.46 ± 0.03 for Field 1 and 〈[Fe/H]〉 = −1.54 ± 0.03 for Field 2, where the errors represent standard errors of the mean. This difference is not statistically significant.

Combining the RR Lyraes in the two fields yields the MDF shown as the solid line in Figure 13, wherein the binned and generalized histograms are shown. The latter has been constructed using an error of 0.31 dex per star (Alcock et al. 2000). The peak metallicity for our sample of ab-type RR Lyraes is then 〈[Fe/H]〉 = −1.50 ± 0.02, where the error is the standard error of the mean. The systematic error of this measurement is likely to be closer to ∼0.3 dex. The dotted distributions in Figure 13 are the binned and generalized histograms for the RRab stars in the sample of B2004 scaled to the same number of ab-type RR Lyraes as in our fields. The peak abundance of this MDF is 〈[Fe/H]〉 = −1.77 ± 0.06.

Zoom In Zoom Out Reset image size

There are three observations we can make regarding the appearance of Figure 13. First, it would seem that the errors on the individual metallicity measurements are significant enough to overwhelm any fine structure that may be present in the two MDFs. That is to say, both MDFs look essentially like normal distributions. Second, since we have applied the same transformation from period to metallicity to both sets of RR Lyraes, we can assert with significant statistical certainty that the mean metal abundance of the RR Lyraes in the B2004 field is lower than that of the RR Lyraes in our two fields. This difference amounts to Δ[Fe/H] = 0.27 ± 0.06 and reflects back on the period shift seen in the Bailey diagram shown in Figure 10. Third, there are a small but non-negligible number of ab-type RR Lyraes with metallicities above [Fe/H] ∼ −1 that are not seen in the B2004 field. Given the fact that our field is closer to the central regions of M31 as compared with the B2004, it is possible that these metal-rich RR Lyraes could belong to the bulge or disk of M31. We return to this point in the following section.

One last point needs to be addressed before leaving this section and that is concerned with the M31 distance implied by the RR Lyraes in our sample. Using the relation advocated by Chaboyer (1999) of M_V(RR) = 0.23[Fe/H] + 0.93 and a reddening of E(B − V) = 0.08 ± 0.03 (Schlegel et al. 1998), we find a distance modulus of (m − M)₀ = 24.46 ± 0.11. This is consistent with the B2004 value and a number of previous determinations.