PROPERTIES OF M31. III. CANDIDATE BEAT CEPHEIDS FROM PS1 PANDROMEDA DATA AND THEIR IMPLICATION ON METALLICITY GRADIENT

Beat Cepheids pulsate simultaneously in two radial modes. Studies of beat Cepheids can be dated back to Oosterhoff (1957a, 1957b), where he introduced a beat period to explain the large scattered photometric measurements of U TrA and TU Cas in the Milky Way. Several attempts to search for Galactic beat Cepheids have been conducted (see, e.g., Pike & Andrews 1979; Henden 1979, 1980), however, only 20 Galactic beat Cepheids have been documented to date (see, e.g., the McMaster Cepheid Data Archive, ⁵ where 651 Type I Cepheids and 209 Type II Cepheids are listed as well). The first larger samples of beat Cepheids have been identified in microlensing survey data. For example, the MACHO project has discovered 45 beat Cepheids in the Large Magellanic Cloud (LMC), where 30 are pulsating in the fundamental mode and first overtone, while 15 are pulsating in the first and second overtone (Alcock et al. 1995). The OGLE team found 93 beat Cepheids in the Small Magellan Clouds (SMCs; Udalski et al. 1999) and 76 beat Cepheids in the Large Magellanic Clouds (Soszynski et al. 2000). A recent study from the EROS group has increased the number of known beat Cepheids in the Magellanic clouds to over 200 (Marquette et al. 2009). The OGLE-III survey has found the largest number of beat Cepheids so far: 271 objects in the LMC (Soszynski et al. 2008) and 277 in the SMC (Soszynski et al. 2010). These numbers will be even larger during the currently conducted OGLE-IV phase.

Beat Cepheids pulsating in the fundamental mode and first overtone can be used as tracers of the metallicity content within a galaxy. This is because, from modeling, there exists only a sub-region in the parameter spaces of mass, luminosity, temperature, and metallicity where both the fundamental mode and first overtone are linearly unstable (see, e.g., Kolláth et al. 2002). Beaulieu et al. (2006) have thus made use of the beat Cepheids found in the CFHT M33 survey (Hartman et al. 2006) and derived the metallicity gradient of M33 to be −0.16 dex kpc⁻¹. Their metallicity gradient supports the H ii region result from Garnett et al. (1997) but disagrees with the much shallower gradient from Crockett et al. (2006), who also used the H ii region to derive the metallicity. It is important to note that the results from both Garnett et al. (1997) and Crockett et al. (2006) are derived from H ii regions, yet are inconsistent with each other.

In this study, we present a sample of beat Cepheids identified from the PS1 PAndromeda project. We derive a metallicity gradient of M31 and compare our results with previous studies of H ii regions and planetary nebulae. Our paper is organized as follows. In Section 2, we demonstrate our method to search for beat Cepheids. We elucidate the approach to derive metallicity in Section 3. The metallicity gradient of M31 from our sample, as well as a comparison with a previous H ii region and planetary nebulae method is presented in Section 4, followed by a conclusion and outlook in Section 5.

We use the optical data taken by the PAndromeda project to search for beat Cepheids. PAndromeda monitors the Andromeda galaxy with the 1.8 m PS1 telescope with a ∼7 deg² field of view (see Kaiser et al. 2010; Hodapp et al. 2004; Tonry & Onaka 2009, for a detailed description of the PS1 system, optical design, and the imager). Observations are taken in r_P1 and i_P1 on a daily basis during July to December in order to search for microlensing events and variables. Several exposures in g_P1, z_P1, and y_P1 are also taken as complementary information for studies on the stellar content.

The data reduction is based on the MDia tool (Koppenhoefer et al. 2013) and is explained in detail in Lee et al. (2012). We outline our data reduction steps as follows. The raw data are detrended by the image processing pipeline (Magnier 2006) and warped to a sky-based image plane (so-called skycells). The images at the skycell stage are further analyzed by our sophisticated imaging subtraction pipeline mupipe (Gössl & Riffeser 2002) based on the idea of image differencing analysis advocated by Alard & Lupton (1998). This includes the creation of deep reference images from best seeing data, stacking of observations within one visit to have a better signal-to-noise ratio (hereafter "visit stacks"), subtraction of visit stacks from the reference images to search for variability, and creating light curves from the subtracted images.

We have shown in Kodric et al. (2013) how to obtain Cepheid light curves in the PAndromeda data. The major difference is that the data set used in this work contains three years of PAndromeda, instead of one year and a few days from the second year data used in Kodric et al. (2013). The sky tessellation is also different, in order to have the central region of M31 in the center of a skycell (skycell 045), instead of at the corner of adjacent skycells (skycell number 065, 066, 077, and 078) as in Kodric et al. (2013); the skycells are larger and overlap in the new tessellation. The new tessellation is drawn in Figure 1. We have extended the analysis to 47 skycells, which is twice as many as were used in Kodric et al. (2013). The skycells we used are 012–017, 022–028, 032–038, 042–048, 052–058, 062–068, and 072–077, which cover the whole of M31. The search of Cepheids is conducted in both r_P1 and i_P1, where we start from the resolved sources in the r_P1 reference images, and require variability in both the r_P1 and i_P1 filters. In addition, one could also search for variables in the pixel-based light curves. This approach would add light curves for fainter variable sources (among them potentially lower period Cepheids) which we do not aim to study in this work.

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We use the SigSpec package (Reegen 2007) to determine the period of all variables. For a given light curve, we iterate the period search five times in both r_P1 and i_P1 to search for multiple periods. In each iteration, SigSpec computes the significance spectrum and determines the most significant period. It then fits a multi-sine function based on this period, subtracts the best-fitted multi-sine curve to the input light curve, and performs another iteration of period search based on this pre-whitened light curve.

For the beat Cepheids, we look for sources that show only two significant periods (i.e., where SigSpec does not find a period after the second iteration). We also require that both periods are found in the r_P1 and i_P1 light curves and are consistent within 10%. We adopt the period derived from r_P1 as the final period, due to the better sampling and the higher amplitude than the i_P1-band light curves. This leads to a sample of 17 beat Cepheids. Their locations, periods in the fundamental mode (P₀) and first overtone (P₁), and light curves are shown in Figures 2, and 3, and Table 1. In the next section, we present their metallicities derived from the period and the period ratio. Given the periods and metallicities, we are also able to obtain an estimate of their ages.

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Table 1. Location and Periods of Our Beat Cepheid Sample

Name	R.A.	Decl.	$P_0 ^{r_{\rm P1}}$	$P_1 ^{r_{\rm P1}}$	$P_1 ^{r_{\rm P1}}$/$P_0 ^{r_{\rm P1}}$	$P_0 ^{i_{\rm P1}}$	$P_1 ^{i_{\rm P1}}$	$P_1 ^{i_{\rm P1}}$/$P_0 ^{i_{\rm P1}}$
	(J2000)	(J2000)	(days)	(days)		(days)	(days)
PSO J010.0031+40.6271	10.00313	40.62716	5.08121 ± 0.00131	3.57641 ± 0.00049	0.703850	5.08163 ± 0.00790	3.57673 ± 0.00161	0.703855
PSO J010.0289+40.6434	10.02891	40.64346	4.42890 ± 0.00084	3.11390 ± 0.00071	0.703087	4.42311 ± 0.00269	3.11339 ± 0.00099	0.703892
PSO J010.0908+40.8632	10.09082	40.86323	3.82737 ± 0.00050	2.69071 ± 0.00028	0.703018	3.82750 ± 0.00103	2.69091 ± 0.01056	0.703046
PSO J010.1097+41.1233	10.10973	41.12339	3.88787 ± 0.00057	2.76180 ± 0.00049	0.710363	3.88655 ± 0.00129	2.76147 ± 0.00076	0.710520
PSO J010.1601+41.0591	10.16016	41.05914	4.81211 ± 0.00068	3.37280 ± 0.00057	0.700898	4.81242 ± 0.00120	3.37283 ± 0.00140	0.700859
PSO J010.2081+40.5311	10.20820	40.53114	3.73484 ± 0.00059	2.66164 ± 0.00085	0.712652	3.73517 ± 0.00191	2.66181 ± 0.01328	0.712634
PSO J010.3333+41.2202	10.33331	41.22027	3.96209 ± 0.00066	2.82765 ± 0.00021	0.713676	3.96449 ± 0.00257	2.82771 ± 0.00043	0.713259
PSO J010.3431+40.8255	10.34310	40.82556	8.68802 ± 0.00299	6.02351 ± 0.00240	0.693312	8.69612 ± 0.00515	6.02638 ± 0.00361	0.692996
PSO J010.5507+40.8208	10.55071	40.82087	4.68226 ± 0.00140	3.28483 ± 0.00052	0.701548	4.67941 ± 0.00300	3.28542 ± 0.00074	0.702101
PSO J010.6214+41.4763	10.62146	41.47634	5.86549 ± 0.00125	4.08473 ± 0.00099	0.696400	5.86891 ± 0.00338	4.08356 ± 0.02249	0.695795
PSO J010.8571+41.7272	10.85714	41.72723	4.12749 ± 0.00071	2.93815 ± 0.00049	0.711849	4.12611 ± 0.00164	2.93745 ± 0.00260	0.711918
PSO J011.2784+41.8935	11.27840	41.89359	4.77328 ± 0.00088	3.37092 ± 0.00056	0.706206	4.77551 ± 0.02909	3.34082 ± 0.01690	0.699573
PSO J011.3670+41.7533	11.36709	41.75335	8.26314 ± 0.00167	5.78110 ± 0.00110	0.699625	8.26723 ± 0.00300	5.78289 ± 0.00169	0.699495
PSO J011.3993+41.6778	11.39932	41.67789	4.81283 ± 0.00192	3.39273 ± 0.00053	0.704935	4.81013 ± 0.00356	3.39258 ± 0.00090	0.705299
PSO J011.4131+42.0052	11.41317	42.00529	3.66634 ± 0.00081	2.60609 ± 0.00043	0.710815	3.66630 ± 0.00272	2.60564 ± 0.00130	0.710700
PSO J011.4436+41.9044	11.44369	41.90446	2.37187 ± 0.00066	1.69231 ± 0.00032	0.713492	2.37164 ± 0.00096	1.69201 ± 0.00750	0.713435
PSO J011.4835+42.1621	11.48356	42.16218	6.09759 ± 0.00176	4.25145 ± 0.00131	0.697234	6.09709 ± 0.00277	4.25231 ± 0.00364	0.697433

Note. r_P1-band columns are shown in bold because we adopt these for the final analysis.

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Given a uniquely measured period (P₀) and period ratio (P₁/P₀) of a beat Cepheid, the pulsation models only allow a sub-region in the parameter spaces of mass, luminosity, temperature, and metallicity for stable double mode pulsations. This enables us to narrow down the metallicity of the beat Cepheids (Beaulieu et al. 2006). As has been shown by Buchler (2008), one can derive the upper and lower metallicity limits simply by the location of a beat Cepheid on the log (P₀) versus P₁/P₀ diagram (the so-called Petersen diagram; Petersen 1973). Buchler & Szabó (2007) and Buchler (2008) showed that the metallicity estimates from this method fall in the generally accepted ballpark for Magellanic Clouds and M33. Figure 4 shows our M31 sample on the Petersen diagram, as well as beat Cepheids from the Milky Way, the Large and Small Magellanic Clouds, and M33 with tracks of different metallicities taken from Buchler (2008). Our sample—similar to the beat Cepheids in the Milky Way—is on the metal-rich side. On the other hand, the beat Cepheids in the Magellanic Clouds appear to be metal-poor.

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We interpolate the theoretical tracks by Buchler (2008) to derive the limits on the metallicity for our beat Cepheids. In Buchler (2008), two different solar mixtures are compared, one from Grevesse & Noels (1993) and the other from Asplund et al. (2005). In this work, we use the metallicity tracks based on the solar mixture of Grevesse & Noels (1993), which agree better with the commonly used values. The metallicity is derived as follows. From the theoretical tracks by Buchler (2008), one can delimit the lower (Z_min; Figure 4, left-hand side) and upper (Z_max; Figure 4, right-hand side) boundaries of a given position in the Petersen diagram by interpolating between isometallicity lines. We adopt the average of Z_min and Z_max as the metallicity estimate Z. The uncertainty is taken as Z_max − Z_min/2. For example, PSO J011.4436+41.9044 has log P₀ ∼ 0.38 and P₁/P₀ ∼ 0.7135. In Figure 4, its lower boundary Z_min is bracketed by isometallicity lines Z = 0.009 and 0.010, while its upper boundary Z_max is between Z = 0.011 and 0.012, as also shown in the zoom-in for the Petersen diagram in Figure 5. By interpolation, we thus derive Z_min = 0.0098 and Z_max = 0.01152. The metallicity estimate is thus Z = (Z_max + Z_min/2) = 0.01066 and the uncertainty is (Z_max − Z_min/2) = 0.00086. The derived metallicity and its error for our sample are shown in Table 2. We also explore the impact on the metallicity estimates from errors in P₁/P₀ and present the results in the Appendix. In Figure 7, when calculating the lower boundary Z_min, we use P₁/P₀ + (error of P₁/P₀) instead of P₁/P₀, and for the upper boundary Z_max, we use P₁/P₀−(error of P₁/P₀). The results are shown in Table 3 in the Appendix, where the metallicity estimates Z remain the same, with or without taking into account the error of P₁/P₀. Only the uncertainty of the metallicity estimates changes very slightly.

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The fact that the uncertainties in the metallicity become large when log (P₀) ∼ 0.84 only allows us to determine the value of Z for 15 out of 17 beat Cepheids in our sample.

Once we have the period and metallicity, we can use the period-age relation from Table 4 of Bono et al. (2005),

$$\begin{equation} \log (t) = \alpha + \beta \log (P), \end{equation} \tag{ 1 }$$

to derive the age of our sample. Here, we use $P_0 ^{r_{\rm P1}}$ to calculate the age. However, one should bear in mind that this period-age relation is for the fundamental mode, but not specially for beat Cepheids. We adopt (α, β) = (8.49, −0.79) for Z < 0.007, (8.41, −0.78) for Z between 0.007 and 0.015, and (8.31, −0.67) for Z between 0.015 and 0.025. The ages of our beat Cepheids are all on the order of ∼100 Myr, showing that they are tracing a rather young stellar population. The age estimates can be found in Table 2.

To derive the metallicity gradient, we first de-project the coordinates of the beat Cepheids to galactocentric distances using the transformation of Haud (1981). We assume that the center of M31 is located at R.A. = 00^h42'4452 (J2000) and decl. = +41^d16'0869 (J2000), with a position angle of 37^d42'54''. We also assume an inclination angle of 125 (Simien et al. 1978) and a distance of 770 kpc (Freedman & Madore 1990).

To compare with previous results from H ii region studies (Zurita & Bresolin 2012; Sanders et al. 2012), which are shown in log(O/H), we first convert our Z values to [O/H] by using a Z_☉ value of 0.017 and [O/H] = [Fe/H]/1.417 from Maciel et al. (2003). We then use log(O/H)_☉ + 12 = 8.69 (Asplund et al. 2009) to calculate the values of log(O/H) + 12 for our sample, and compare them with previous results from the H ii regions and planetary nebulae observations of M31 shown in Figure 6.

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There are several ways to extract the chemical abundance from spectroscopic observations of H ii regions. For example, Zurita & Bresolin (2012) have determined the electron temperature of the gas from H ii regions and derive the chemical abundance accordingly (so-called direct-T_e method). On the other hand, one can use the flux ratio between strong lines to infer the chemical abundance of certain elements. For example, Sanders et al. (2012) have used the flux ratio between [N ii] and H_α proposed by Nagao et al. (2006) to obtain the log(O/H) values from H ii regions. In Figures 2 and 6, we only show the eight H ii region samples from Zurita & Bresolin (2012), because they are the only ones who derive T_e and [O/H] values from the faint [O iii] line directly. In Sanders et al. (2012), they have hundreds of H ii region measurements, but their [O/H] value varies depending on which strong lines are used. Our beat Cepheid result is closer to the metallicities from the direct method of Zurita & Bresolin (2012) than the strong line mentioned. Our errors are much smaller than those for traditional metallicity measurement methods. As a consequence, the difference of our metallicity values from those of Zurita & Bresolin (2012) is significant.

In addition to the H ii regions, chemical abundance can be derived from the planetary nebulae as well. We also compare our result to the metallicities from Kwitter et al. (2012) in Figure 6. Contrary to the metallicities from planetary nebulae, our result shows a sub-solar log(O/H) value within 15 kpc, similar to the result from H ii regions. The mean log(O/H)+12 value from our sample is 8.56, while observations from planetary nebulae give a higher value (8.64). Our sample has a gradient of −0.008 ± 0.004 dex kpc⁻¹, which is close to the value of −0.011 ± 0.004 dex kpc⁻¹ from planetary nebulae (Kwitter et al. 2012). Our result shows scatter around the linear gradient, which could originate from the intrinsic variation of in situ metallicity.

The detailed properties of our sample, including the metallicity, galactocentric distance, and age, are shown in Table 2.

We present a sample of the beat Cepheids based on the PAndromeda data. We use the P₁/P₀ to P₀ relations from pulsation models of Buchler (2008) to estimate the Cepheid metallicities. We de-project the location of beat Cepheids, and derive the metallicity gradient of M31. Our result is closer to the results from the planetary nebulae of Kwitter et al. (2012).

In this work, we only concentrate on searching beat Cepheids from a sample of resolved sources. In a future work, we will also conduct searches for variables from pixel-based light curves. In this case, we could find fainter variables.

Because the beat Cepheids are pulsating at relatively short periods, they are intrinsically very faint, and with a 2 m class telescope like PS1 it is difficult to find a large sample at the distance of M31. To increase the number of beat Cepheids in M31, deeper surveys are required. Our understanding of beat Cepheid content in M31 can be improved with the CFHT POMME survey (Fliri & Valls-Gabaud 2012) and the upcoming LSST project (Ivezic et al. 2008).

We acknowledge comments from the anonymous referee. We are grateful to Andrzej Udalski for his useful comments. This work was supported by the DFG cluster of excellence "Origin and Structure of the Universe" (www.universe-cluster.de). The Pan-STARRS1 Surveys (PS1) have been made possible through contributions from the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under grant No. AST-1238877, and the University of Maryland.

In this section, we explore the impact on the metallicity estimates from errors in P₁/P₀ and present the results. In Figure 7, when calculating lower boundary Z_min, we use P₁/P₀ + (error of P₁/P₀) instead of P₁/P₀, and for the upper boundary Z_max, we use P₁/P₀−(error of P₁/P₀). The results are shown in Table 3, where the metallicity estimates Z remain the same, with or without taking into account the error of P₁/P₀. Only the uncertainty of the metallicity estimates changes very slightly.

Table 3.  Z of Beat Cepheid Properties; Derived with and without Taking into Account Errors in P₁/P₀

Name	Z	Z
	without $\frac{P_1}{P_0}$-err	with $\frac{P_1}{P_0}$-err
J010.0031+40.6271	0.0116 ± 0.0011	0.0116 ± 0.0012
J010.0289+40.6434	0.0129 ± 0.0012	0.0129 ± 0.0014
J010.0908+40.8632	0.0140 ± 0.0014	0.0140 ± 0.0014
J010.1097+41.1233	0.0091 ± 0.0009	0.0091 ± 0.0010
J010.1601+41.0591	0.0139 ± 0.0013	0.0139 ± 0.0014
J010.2081+40.5311	0.0081 ± 0.0008	0.0081 ± 0.0010
J010.3333+41.2202	0.0073 ± 0.0008	0.0073 ± 0.0008
J010.5507+40.8208	0.0136 ± 0.0013	0.0136 ± 0.0015
J010.6214+41.4763	0.0158 ± 0.0015	0.0158 ± 0.0016
J010.8571+41.7272	0.0081 ± 0.0008	0.0081 ± 0.0009
J011.2784+41.8935	0.0105 ± 0.0009	0.0105 ± 0.0011
J011.3993+41.6778	0.0112 ± 0.0010	0.0112 ± 0.0012
J011.4131+42.0052	0.0092 ± 0.0009	0.0092 ± 0.0010
J011.4436+41.9044	0.0107 ± 0.0009	0.0107 ± 0.0010
J011.4835+42.1621	0.0149 ± 0.0013	0.0149 ± 0.0015

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