PANCHROMATIC HUBBLE ANDROMEDA TREASURY. XIV. THE PERIOD–AGE RELATIONSHIP OF CEPHEID VARIABLES IN M31 STAR CLUSTERS

We select Cepheids from the Pan-STARRS Cepheid sample presented by Kodric et al. (2013), which is large and well-characterized. Observations were made in the ${i}_{{\rm{P}}1}$ and ${r}_{{\rm{P}}1}$ filters over 183 epochs from 2010 July 23 to 2011 August 12 (with a half-year gap in the schedule). Before inclusion in the final sample, the candidates passed cuts based on CMD location and light-curve Fourier parameters. Kodric et al. manually classified a subset of the light curves and defined regions in Fourier space corresponding to fundamental mode, first-overtone, and Type-II Cepheids based on these classifications. This enabled them to automatically classify all of the objects in their catalog.

The PS1 light curves are well-sampled, allowing us to extend them to the epoch of HST observation with minimal error. We can thus correct for the phase of the Cepheid at the times of our HST observations, and derive accurate mean magnitudes from even the small number of HST observations. The process by which we extend the PS1 light curves is described in Section 3.

The most reliable way to assign ages to Cepheids is to associate them with a single age stellar population. At the distance of M31, the main-sequence turnoff for even young populations is near the HST detection limit. Though it is possible to assign ages to recent bursts of star formation in any region of the M31 disk (e.g., Jennings et al. 2012), the results suffer from potential bias due to the presence of overlapping populations and small numbers of coeval stars. We are interested in robustly testing the period–age relationship, and bias in the age estimates (especially from older underlying populations) could significantly affect the result. Thus, we focus on Cepheids near high-confidence clusters, where we can expect to obtain strong age constraints.

The Andromeda Project (hereafter AP) provides an M31 cluster sample of unprecedented size and quality, derived from the PHAT survey (Johnson et al. 2015). The PHAT data consists of HST Wide Field Camera 3 (WFC3) and Advanced Camera for Surveys (ACS) imaging in 6 filters covering $\sim 0.5\;{\mathrm{deg}}^{2}$ of the northeast M31 disk (Dalcanton et al. 2012). The AP clusters were identified through visual inspection of optical ACS/WFC F475W and F814W search images by citizen scientist volunteers. We use the final AP cluster catalog of 2753 clusters for this work.

To form our sample of candidate cluster Cepheids, we first search for PAndromeda Cepheids whose reported positions place them within the extent of an AP cluster aperture. Since the AP cluster radii and centers are somewhat uncertain, we first define the extent generously by expanding the AP radius by a factor of 1.2; however, all matches found were within the nominal cluster radii. This search yields 9 fundamental mode (FM) Cepheids and 2 left unidentified (UN) by Kodric et al. (2013). Fundamental, first-overtone, and Type-II Cepheids must be studied independently for precise period–age relations; but since all but two of our objects are firmly identified as FM pulsators and our sample size is already small, we proceed under the assumption that they are all FM Cepheids. Table 1 details the basic properties of this set. The spatial and PL distribution of the sample is depicted in Figures 1 and 2. In addition, we note that all of the clusters selected have final weighted cluster fractions (discussed in Johnson et al. 2015) f_clst,W > 0.6, suggesting that all of these Cepheids fall within highly probable clusters.

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Table 1.  PHAT/PS1 Cluster Cepheids

CC ID	AP ID	PS ID	ClusterFrac	R.A. (J2000)	decl. (J2000)	Mode	Period, rP1 (days)	${i}_{{\rm{P}}1,\;{HST}}-{i}_{{\rm{P}}1,\;\mathrm{PS}1}$
1	477	PSO-J011.4286 + 41.9275	0.9881	11.42865	41.92759	FM	4.582	1.19 ± 0.08
2	1539	PSO-J011.4279 + 41.8642	0.9358	11.42797	41.86422	UN	6.08	0.99 ± 0.21
3	3928	PSO-J011.6227 + 41.9637	0.8738	11.62273	41.96373	FM	6.213	0.09 ± 0.06
4	2831	PSO-J011.6374 + 42.1393	0.9059	11.63742	42.13933	UN	7.928	0.88 ± 0.05
5	3050	PSO-J010.9769 + 41.6171	0.8461	10.97692	41.61718	FM	8.829	0.20 ± 0.10
6	5216	PSO-J011.0696 + 41.8571	0.7049	11.0696	41.85714	FM	14.353	0.48 ± 0.22
7	2113	PSO-J011.6519 + 42.1286	0.9807	11.65193	42.12865	FM	15.429	0.11 ± 0.13
8	2587	PSO-J011.7119 + 42.0449	0.8284	11.71191	42.04491	FM	17.18	0.29 ± 0.10
9	2967	PSO-J011.0209 + 41.3162	0.6609	11.02092	41.31625	FM	19.566	0.65 ± 0.10
10	1082	PSO-J011.2797 + 41.6217	0.9087	11.27976	41.62174	FM	26.499	0.31 ± 0.03
11	1540	PSO-J011.3077 + 41.8500	0.984	11.30773	41.85003	FM	35.75	0.24 ± 0.06

Note. Table 1 lists the Cluster Cepheids (CCs) identified from the PS1/PHAT data, and provides some basic information about each. The AP ID identifies the cluster in the Johnson et al. (2015) catalog and f_{clst, W} represents the final cluster quality parameter discussed there. The PS ID identifies the Cepheid from the PS catalog. The period in the ${r}_{{\rm{P}}1}$ filter; Cepheid R.A. and decl.; the mode classification (FM for fundamental mode, UN for unidentifed) determined by Kodric et al. (2013) are also provided. Finally, the last two columns indicate the HST-determined correction to the PS1 mean magnitudes, as discussed in Section 3.2.

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The PHAT survey implementation can be exploited to extract multi-epoch data for most points in the survey grid. This is due to the fact that the HST tiling is based on WFC3/IR, which has a substantially smaller field of view than ACS/WFC; so simultaneous pointings of these cameras generate large overlap in the ACS images. This means that for all but one of the Cepheids we have deep HST imagery from at least two different times, and thus a means to assess variability from the HST data alone. This extra information can be used to distinguish the Cepheid from its neighbors at very high confidence. The number of HST images available for the clusters under consideration ranges from 1 to 4 with a mean of 2.7.

We first attempt to identify each Cepheid as a resolved source in the HST photometry. Spatial agreement is checked by cluster proximity. As the cluster fields are crowded and the PS-1 determined positions are given to an accuracy of only 036, we consider all stars within the AP cluster radius as candidates. We consider two additional pieces of evidence when cross matching the Cepheids; location on the CMD and variability.

Proximity to the instability strip. We adopt the "non-canonical," solar-metallicity instability strip boundaries derived by Bono et al. (2005) for fundamental pulsators satisfying $0.17\lt {\mathrm{log}}_{10}(\mathrm{period}/\mathrm{days})\lt 2.02.$ In order to transform the boundaries from theoretical ${T}_{\mathrm{eff}}$-Luminosity space to HST filter space, we use the MATCH (Dolphin 2002) fake utility to sample the Padova isochrones (discussed in Section 3.3) at various ages and thereby populate the instability strip with model stars. The model stars which fall between the blue and red edges and within the valid period range (determined using the pulsational relations derived in Bono et al. 2000) are then plotted in HST F475W-F814W, F814W color–magnitude space. The quadrilateral which most-tightly encloses these points in CMD space defines our observational instability strip. Indeed we have used the transformations from Sirianni et al. (2005), as described in Girardi et al. (2008; as these transformations are used by MATCH to yield filter magnitudes).

Membership in the final filter-transformed instability strip is uncertain due to errors in the reddening adopted for each cluster. We estimate A_V by fitting the CMD of the full cluster (as described in Section 3.3). The uncertainty in the reddening estimates from these fits is on the order of 0.1 mag. To ensure that we catch all reasonable candidates, we expand the edges of the strip to cover an A_V range of ±1.5 mag around the best fit, as illustrated in Figure 3. This is a very large overestimate intended to prevent accumulated error from the HST photometry, the filter corrections, and pulsational models from excluding too many objects from consideration.

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Variability. For each cluster, we identify all ACS observations from the PHAT survey whose borders contain the cluster. We select a reference field in which the cluster is fully contained and that is not too close to the chip gap (where distortion potential is highest). We start with the st photometry catalogs (described by Williams et al. 2014), and apply an additional cut in signal-to-noise ratio (greater than 4 in both filters). These catalogs are produced by the PHAT reduction pipeline, and do not include quality cuts on PSF shape or crowding; we wish to maximize catalog completeness at the expense of photometry quality. In order to match the stars in different overlapping observations, we use pattern-matching code from Groth (1986) to determine R.A./decl. offsets from the brightest stars in each field. For each cataloged source in the reference field, we search a circle of radius .072 arcsec in the other images for matches. This radius is on the order of the FWHM for ACS, and our results indicate that it matches all sources of interest. If more than one source is found in any of these circles, we consider it a non-detection in the corresponding field. The sources for which we obtain at least two exposures are retained for analysis. We compare the resulting catalog to the full st reference field catalog in the instability strip region to ensure that no candidates are missed due to alignment issues.

Before we can compare the PHAT and PS1 data, we must extend the PS1 light curves to the HST observation epochs. For each Cepheid in their final sample, Kodric et al. (2013) provide the raw unfolded light curves and the coefficients of a fifth-order Fourier series fit to the folded light curves. Unfortunately, they do not appear to provide a zero-epoch for these determinations. We use a simple scalar minimization routine to determine the zero epoch which leads to the best agreement between the Fourier series and the data. The result is checked by-eye, and the uncertainty in this epoch determination is computed by bootstrap resampling. This procedure allows us to compute the Cepheid magnitude expected by PS1 at an arbitrary epoch. Uncertainties in the PS1 expected magnitudes are propagated by Monte Carlo methods, assuming that the parameters and associated errors provided by Kodric represent Gaussian distributions.

We compute several variability indices inspired by Welch & Stetson (1993) for each matched source. These quantify the degree to which the magnitudes in two filters are correlated from field to field. They are calculated for our pair of HST ACS magnitudes F475W and F814W for each source, as well as for the source F814W and the model expected ${i}_{{\rm{P}}1}$ Cepheid magnitude. We denote filter 1, filter 2 magnitude in the kth field by $F{1}_{k},F{2}_{k}$, respectively, the corresponding pipeline-determined Poisson uncertainty in each by ${\sigma }_{f1,k},{\sigma }_{f2,k},$ and the variance-weighted mean for each filter by $\bar{F1}.$ We then define the variability index I to be

$$\begin{eqnarray*}&&I=\displaystyle \frac{1}{\sqrt{n(n-1)}}\displaystyle \sum _{i=1}^{n}\left(\displaystyle \frac{F{1}_{k}-\bar{F1}}{{\sigma }_{f1,k}}\right)\left(\displaystyle \frac{F{2}_{k}-\bar{F2}}{{\sigma }_{f2,k}}\right),\end{eqnarray*}$$

where n is the number of fields in which the source was matched. We denote the index computed with the 2 ACS filters as ${I}_{\mathrm{ACS}}$, and with the F814W and expected ${i}_{{\rm{P}}1}$ as ${I}_{\mathrm{ACS}-\mathrm{PS}1}$. Note that this index is the covariance of the measurements normalized by the photometric uncertainty. For a truly non-variable source with good photometry and good background subtraction, the variation in measured brightness in two filters should be independent. Thus the expectation value of I for non-variable stars with ideal photometry is zero. A higher value of I indicates a higher degree of correlation between brightness changes in the two filters. A high value of the index may indicate a truly variable source; but it can also reflect issues with photometry, background subtraction, or source matching. Here, our situation is simplified—we expect that there is a variable within the field being examined, and the indices are used to help us identify which candidate resolved source it is most likely to be.

We select as candidates the sources whose weighted-average ACS magnitudes place them inside of the generous instability strip for the cluster (yellow-outlined region in Figure 3). We identify the putative Cepheid as the candidate source with the maximum variability index, considering both filters equally. Since the photometric errors of the ACS observations are substantially smaller than those assigned to the PS1 magnitudes by our Monte Carlo uncertainty estimation, the ACS variability index usually dominates (especially when the brightness changes are slight). This selection process is illustrated for each cluster in Figure 3. For all clusters except AP5216, we find a clear variability standout in the expected region of the CMD. Since AP5216 was observed in only one PHAT field, no variability information is available for the HST sources; fortunately, the generous instability strip encloses exactly one cluster source, which we identify as CC6.

We now proceed to compare the HST and PS1 photometry of the Cepheids. The cluster environments are crowded, so we expect the PS1 measurements to be systematically brighter due to source confusion. With our Cepheids identified in the HST data, we can test the actual size of this effect. In this section, we use the absolute flux estimate from the HST photometry to re-estimate the Cepheid mean magnitudes. This requires that we compare the HST photometry to the PS1 models in a common filter system.

Since we are only concerned with Cepheids, whose SEDs are fairly homogeneous and not unusual, a standard color-based transformation is appropriate here. Both the F814W ACS filter and the ${i}_{{\rm{P}}1}$ filter are close to the Landolt (Johnsons/Cousins) I. Tonry et al. (2012) has conducted an analysis of the Pan-STARRS1 photometric system, which includes transformations to the Johnsons/Cousins system derived from a library of 783 photometric-standard SEDs. Similarly, Sirianni et al. (2005) presents transformations from ACS magnitudes to Johnsons/Cousins based on synthetic magnitudes and checked with empirical data. We opt to use the quadratic relations in both cases. After de-reddening the HST data using our cluster A_V, we transform each pair of F475W/F814W magnitudes to V/I by iteratively improving the V − I color term. The PS1 transformations require we use B − V color. Since we are missing a coincident B observation, we opt to estimate the Cepheid's average B − V color using an empirical period-color relation from Tammann et al. (2003) for galactic Cepheids. To account for the variation of B − V over the period of the Cepheid, we add 0.2 mag in-quadrature to the color uncertainty. The reddening is re-applied to the final magnitudes using the prescriptions from Tonry et al. (2012). So for each field in which the Cepheid candidate appeared, we obtain an HST-derived estimate of the Cepheid brightness in the ${i}_{{\rm{P}}1}$ filter at that field epoch, which we can compare to the PS1 model magnitude.

Now we use the HST data to cross-calibrate the PS1 light curves. The PS1 magnitude variation provides a good estimate of the true change in flux received from the Cepheid, but the absolute level of the flux is better determined by HST. At each HST epoch, we compute the difference in zeropoint-scaled flux between the PS1-determined mean magnitude and the PS1 model magnitude at that time:

$$\begin{eqnarray*}&&{\rm{\Delta }}f={10}^{-0.4{\bar{i}}_{P1}}-{10}^{-0.4{i}_{{\rm{P}}1,\mathrm{model}}}\end{eqnarray*}$$

then add this to the HST-determined magnitude in the ${i}_{{\rm{P}}1}$ filter:

$$\begin{eqnarray*}&&{\bar{i}}_{{\rm{P}}1,{HST}}=-2.5{\mathrm{log}}_{10}\left({10}^{-2{i}_{{\rm{P}}1,{\text{}}{HST}}/5}+{\rm{\Delta }}f\right).\end{eqnarray*}$$

We repeat this for each field, and propagate the uncertainties via Monte Carlo methods (assuming Gaussian distributions for the parameters). Thus, for each Cepheid, we obtain at least 2 estimates of a calibrated mean magnitude in i_P1.

The results of this dephasing step are shown in Figure 4. We plot the difference between the HST-calibrated mean magnitude and the PS1 mean magnitude (averaging the results for all PHAT fields the star was located in), against a blending measure. The measure is computed by summing the F814W flux of all .dst catalog sources within a radius of 086 (3.3 parsecs in the disk) of the putative Cepheid. In describing the PAndromeda data pipeline, Lee et al. (2012) reports for one representative skycell a median FWHM in the 30 best images of 0861. Thus, the metric represents a rough estimate of the degree to which blending affects the PS1 data. The HST-calibrated means are systematically dimmer than the PS1 means, and there is a trend toward better agreement with less excess (HST-cataloged) flux near the Cepheid.

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As expected in crowded cluster environments, the PS1 magnitudes appear to be systematically brightened by blending. This effect has been studied in-depth previously, e.g., in relation to the DIRECT project by Mochejska et al. (2000) and by Vilardell et al. (2007). In finding a PL relation for their sample, Kodric et al. (2013) performs iterative 3σ clipping to eliminate blends and other outliers. We repeat their procedure, with both the full set of Kodric FM and UN Cepheids; and with the FM Cepheids alone. In both cases, only one Cepheid from our cluster Cepheids sample (CC1) is cut (after 5 iterations in the former, and on the first iteration in the latter case). This leaves 2 (4, including the UN) Cepheids in the PL-determining set whose PS1 measured magnitudes are offset at the 0.5–1.0 mag level.

We also find the Cepheid magnitudes in the near-infrared filter F160W. The effects of random-phase observations are diminished substantially in this filter, as exploited in M31 by Riess et al. (2012), Kodric et al. (2015), and Wagner-Kaiser et al. (2015). The F160W magnitudes of eight of our cluster Cepheids (three were not imaged in NIR filters) are plotted in Figure 5. The empirical (linear) F160W PL relation derived by Wagner-Kaiser et al. is also plotted. Our Cepheids show good agreement with this relationship within the measured uncertainty.

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We estimate the age of the host stellar clusters by fitting populated isochrones to their CMDs. In particular, we use Padova isochrones and assume a Kroupa IMF. These isochrones combine the tracks from Girardi et al. (2000) through the first thermal pulse on the AGB with tracks that treat the TP-AGB in better physical detail, as described in Marigo et al. (2008). Corrections from Girardi et al. (2010) concerning the AGB are also incorporated. In this paper, we are concerned with young Cepheid-hosting clusters only, so the TP-AGB/AGB corrections are not critical. The Padova isochrones are computed over a large age range (from 6.60 to 10.10 dex in $\mathrm{log}(t/\mathrm{years})$) which should comfortably capture all FM Cepheids. The Padova models use a moderate amount of convective overshoot.

We fit the cluster CMDs using MATCH (Dolphin 2002). We run MATCH in a mode that fits an SSP to the cluster CMD. MATCH generates model CMDs which include the effects of cluster-specific artificial star tests, and thus has some defense against spurious photometry effects in these crowded fields. The methodology, fitting parameters, and data products (photometry, artificial star tests, etc.) are the same as those used in the (forthcoming) CMD analysis of the full PHAT cluster sample (L. Beerman et al. 2015, in preparation). MATCH accounts for potential field star contamination in the cluster CMD by incorporating a smooth binning of likely field stars situated in an annulus 10 times the size of the cluster aperture in the fit. We fix the cluster distance modulus at 24.47, and allow the metallicity of the cluster to vary from ${\mathrm{log}}_{10}({Z}_{\mathrm{min}})=-0.20$ to ${\mathrm{log}}_{10}({Z}_{\mathrm{max}})=0.10$ in steps of 0.1 dex. The runs are performed with a resolution in ${\mathrm{log}}_{10}(t/\mathrm{years})$ of 0.10 over the range (6.6, 10.1); and a resolution in A_V of 0.05 over the range (0.00, 3.00). These ranges encompass all physically reasonable solutions, assuming the Cepheids are cluster members. By fitting all models in this grid, and marginalizing over A_V and metallicity, we obtain a posterior probability density function (PDF) for the cluster age assuming the model we have adopted.

We perform the fits with and without the Cepheid included in the data. This serves several purposes. First, isochrone fits are most robust (and degeneracy is minimized) when they are determined by main-sequence stars rather than a handful of bright evolved stars. This is especially worth considering in these cases where the Cepheid is often the only star in the corresponding section of the CMD. Second, this enables us to perform a meaningful test of the period–age relation. If the CMD fits are controlled by the Cepheid photometry, then the resulting MATCH age estimate is effectively a measurement of the Padova isochrone age at a given blue loop magnitude. Finally, since we are not certain that the Cepheid belongs to a dominant SSP as we have assumed, we can look for large changes in the PDF when the Cepheid is removed, which might indicate that this assumption fails. The reported ages used in Section 4 are those derived from the CMDs without the Cepheid.

In most cases, the PDFs with and without the Cepheid are very similar. Six clusters display differences in the best-fit age at our 0.10 dex resolution, and three of these show differences in the median age estimates. We can produce a rough approximation of the standard error of these estimates from half the 84–16 percentile difference. Only one cluster (CC2) shows a difference in both best-fit and median age which exceeds the joint errors. The actual shift in both age estimates is only 0.20 dex. Several other PDFs show hints of a bimodal shift when the Cepheid is removed. Qualitatively, the lack of large shifts in the fitting results indicate that the MATCH fits are indeed capturing an underlying SSP which is roughly consistent with the Cepheids.

The likelihood of the best-fit SSP model relative to the null model (no SSP, i.e., based only on the empirical field star CMD) is another test of fit quality. We re-run MATCH on each cluster, but constrain it to use only the smoothed background model derived from the cluster annulus. We are concerned with the quantity ${\rm{\Delta }}f={f}_{\mathrm{best}}-{f}_{\mathrm{background}}$, where f is the fit value MATCH returns. This f is a Poisson-statistics analog to ${\chi }^{2},$ and thus roughly corresponds to $-2\mathrm{ln}(P);$ so the quantity ${e}^{-{\rm{\Delta }}f/2}\sim {P}_{\mathrm{best}}/{P}_{\mathrm{background}}$ informs us how much more likely the best fit found is relative to the null background fit. This likelihood ratio ranges from 10³ to 10²⁸ for our cluster fits without the Cepheid. The outliers at the low end of this quantity have correspondingly broad normalized PDFs, by necessity. We conclude that there are no catastrophic failures in the fitting results, and proceed with the full cluster age PDFs (derived without the Cepheids) in hand.

We adopt a probabilistic approach to fitting our data. This enables us to properly incorporate the full (non-Gaussian) probability distributions we computed using MATCH for the cluster ages, and to account for errors in the period determination from Kodric et al. (2013; substantial in two cases, as shown in Figure 2).

We fit a linear model to the period and age data: ${\mathrm{log}}_{10}(\mathrm{age}/\mathrm{years})=m{\mathrm{log}}_{10}(\mathrm{period}/\mathrm{days})+b,$ with Gaussian scatter σ in magnitude about the line. Rather than fit with m directly, we use $\theta =\mathrm{arctan}(m)$ so that a flat prior doesn't favor steeper slopes. Given our data (represented by D) and our SSP model for the clusters (Π), we would like to determine the probability function for a linear slope and intercept: ${\mathcal{P}}(\left\{\theta ,b,\sigma \right\}\;| \;D,{\rm{\Pi }}).$ Baye's rule informs us that

$$\begin{eqnarray*}&&{\mathcal{P}}(\left\{\theta ,b,\sigma \right\}\;| \;D,{\rm{\Pi }})\propto {\mathcal{P}}(D\;| \;\left\{\theta ,b,\sigma \right\},{\rm{\Pi }})\ {\mathcal{P}}(\left\{\theta ,b,\sigma \right\}),\end{eqnarray*}$$

where ${\mathcal{P}}(\left\{\theta ,b,\sigma \right\})$ represents our prior expectation for the fit parameters. We adopt a very generous prior (flat over $(-\pi /2\lt \theta \lt 0)$ and all possible intercepts). To evaluate the likelihood function ${\mathcal{P}}(D\;| \;\left\{\theta ,b,\sigma \right\},{\rm{\Pi }}),$ we integrate the product of (1) the MATCH-provided PDFs, (2) the Gaussian period PDFs, and (3) the modeled Gaussian variance about the line (θ, b, σ) over a grid of points for each Cepheid. The product of these likelihoods for all the Cepheids yields the full likelihood function. We apply Markov chain Monte Carlo (MCMC; using the package emcee: Foreman-Mackey et al. 2013) to sample the resultant posterior distribution.

The MCMC fit is displayed in Figure 6. Using the 16th/50th/84th percentiles of the MCMC chain, we obtain estimates for the slope and intercept of $m=-{0.69}_{-0.20}^{+0.25}$ and $b={8.38}_{-0.36}^{+0.37},$ and for the scatter $\sigma ={0.25}_{-0.20}^{+0.33}.$ The constraints are broad, as can be seen in the wide range of the sampled lines in Figure 6. This is a product of the broad age PDFs, which in turn indicate the difficulty of assigning an age to low-mass clusters at the distance of M31.

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We compare our results to several period–age relations in the literature. It is important to note that the choice of evolutionary model used (in fitting clusters or modeling Cepheids) will in general affect the period–age relation, as it affects the age–luminosity relation for stars near the instability strip. From Bono et al. (2005), we compare with the classical, fundamental-mode, relation, for Z = 0.02 (solar metallity). This relation was derived using stellar models from Pietrinferni et al. (2004) which do not include convective core-overshooting.

It is more difficult to find period–age relations derived using models which incorporate overshooting, such as the Padova models we used for CMD fitting. Thus, we derive our own semi-empirical period–age relation from the Padova models. Specifically, we use the MATCH utility fake to sample the Padova isochrones in the range $6.6\lt \mathrm{log}(\mathrm{age}/\mathrm{years})\lt 8.5,$ at a resolution of 0.05 dex. We adopt a sufficiently high star formation rate to populate all instability strip crossings. All other parameters (except initial distance modulus) are kept identical to their state in the CMD fitting (Section 3.3). We select the models lying within the T_eff, luminosity boundaries derived by Bono et al. (2005; solar metallicity, non-canonical). We then utilize the theoretical pulsation relations for fundamental Cepheids derived by Bono et al. (2000) to relate the model temperatures, luminosities, and masses to pulsational periods. Then a simple linear fit is performed to the resulting points in period–age space, providing the desired theoretical prediction.

For further comparison, we locate two additional relations from the literature. Magnier et al. (1997) calculate a period–age relation in a semi-empirical way: they use tracks from Schaller et al. (1992) which allow for overshooting, and an empirical PL relation found for M31 Cepheids. Efremov (2003) fit a period–age relation empirically to a sample of Cepheids from the LMC bar, utilizing integrated color results based upon models from Bertelli et al. (1994) which do incorporate moderate convective core overshooting; we quote their most probable relation. We plot these period–age relations in Figure 6 alongside our MCMC fits, and compare the fit parameters in Table 2. The median estimates agree reasonably well with the ensemble of other values given the large uncertainties in our fit.

Table 2.  Period–Age Relations ${\mathrm{log}}_{10}(\mathrm{age}/\mathrm{years})=m{\mathrm{log}}_{10}(\mathrm{Period}/\mathrm{days})+b$

Source	m	b	Scattera
M31 CC fit	$-{0.69}_{-0.20}^{+0.25}$	${8.38}_{-0.36}^{+0.37}$	${0.25}_{-0.20}^{+0.33}$
Padova fit	$-0.53$	8.40	0.07
Bono et al. (2005)	$-0.67$	8.31	0.08
Magnier et al. (1997)	$-0.6$	8.4	⋯
Efremov (2003)	$-0.65$	8.50	⋯

Note.

^aPredicted scatter in magnitudes about the best-fit relation.

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It is interesting to note that the largest disagreement in slope value is between our empirical fit and the Padova model predictions we derived, despite the fact that both rely on the same set of stellar models. To check whether this discrepancy is due to our choice of pulsational models, we also applied the approach of Magnier et al. (1997) and used empirical PL relations to relate the theoretical model luminosities to pulsational periods and re-derive a Padova period–age prediction. The MATCH fake utility provides magnitudes in common filter systems for the model stars (Girardi et al. 2008). We re-derive the period–age relation using several different PL relations: two from Wagner-Kaiser et al. (2015) for the NIR HST filter F160W and for the I band; and two from Fouqué et al. (2007) derived using galactic Cepheids in the V and I bands. The slopes derived using Wagner-Kaiser et al. (2015) are close to the shallow theoretical Padova results described above: m ≃ −0.55 for both F160W and I. In contrast, the slopes found using Fouqué et al. (2007) are closer to our steeper empirical fit: m ≃ −0.78, −0.69 for V, I, respectively. We expect that the results from Wagner-Kaiser et al. (2015) will be more robust, since their sample size is larger and Cepheid variability is reduced at long wavelengths. This reinforces the suggestion from the theoretical Padova fit that we should expect a shallower slope. Confirming this experimentally would require further testing on clusters with better constraints than our M31 sample can provide.

We conclude that our Cepheid age measurements are in broad agreement with existing theoretical and empirical period–age relations. There are several important caveats to note. Comparing ages derived using stellar models with differing assumptions about overshooting is not strictly valid, as overshooting parameters have an effect on the derived age at fixed luminosity. In addition, we have compared period–age relations derived at different metallicities (solar and $\mathrm{LMC}/\mathrm{SMC}$). Our constraints on the period–age relation are too loose to distinguish these second-order effects; but they must be recognized when applying these tools to derive age estimates.