CEPHEID PERIOD–LUMINOSITY RELATIONS IN THE NEAR-INFRARED AND THE DISTANCE TO M31 FROM THE HUBBLE SPACE TELESCOPE WIDE FIELD CAMERA 3^*

M31, the nearest analogue of the Milky Way Galaxy, has long provided important clues to understanding the scale of the universe. The naked-eye visibility in 1885 of a supernova in M31 (S And, see De Vaucouleurs & Corwin 1985) once suggested that the spiral nebulae were located within the Milky Way lest its luminosity be "on a scale of magnitude such as the imagination recoils from contemplating" (Clerke 1903). Hubble's (1929) subsequent discovery of Cepheid variables in M31 revealed the true gulf that existed between the Milky Way and other galaxies. The ability to resolve the stellar populations of M31 and knowledge of its distance still provide the best constraints on the timescales of the formation of such massive galaxies (Brown et al. 2006). Despite long past success in identifying Cepheids in M31 (Baade & Swope 1963, 1965), comprehensive inventories of its variables were not completed until the availability of wide-format CCD arrays to survey its 3° span and image-subtraction techniques to contend with the significant crowding of its stars, a consequence of its 77° inclination. Large-scale variability surveys in the 1990s by Magnier et al. (1997), the DIRECT Program (e.g., Bonanos et al. 2003), and Vilardell et al. (2007), as well as microlensing surveys in the 2000s, by POINT-AGAPE (An et al. 2004) and WeCAPP (Fliri et al. 2006), succeeded in discovering ∼10³ Cepheids, most from >50 Myr old supergiants at short periods (P < 10 days). The Pixel Observations of M31 with MEgacam (POMME) Survey has been the most prolific to date, making use of Megacam on Canada–France–Hawaii Telescope (CFHT) to discover more than 2500 Cepheids (J. Fliri et al. 2012, to be submitted).

The utility of Cepheids as distance indicators critically depends on the accuracy of their measured fluxes. Even the best optical measurements of M31 Cepheids from the ground have been biased bright at the 0.1–0.2 mag level by stellar crowding (Mochejska et al. 2000; Vilardell et al. 2007) and contaminated by variable extinction owing to the high inclination of its host. Space-based observations in the near-infrared have the ability to greatly mitigate both of these sources of error. A modest sample of eight Cepheids in M31 was measured with greater resolution and in the near-infrared using the NICMOS Camera on the Hubble Space Telescope (HST) by Macri et al. (2001), suggesting that both crowding and extinction could be reduced by such measurements.

The Panchromatic Hubble Andromeda Treasury (PHAT) Program (PI: J. Dalcanton) is an HST Multi-cycle Treasury Program to map roughly a third of M31's star-forming disk, using six filters covering from the ultraviolet through the near-infrared. With HST's resolution and sensitivity, the disk of M31 will be resolved into more than 100 million stars, enabling a wide range of scientific endeavors. Despite capturing only a static view of M31, the random phase observations of known Cepheids obtained by PHAT are nearly as precise as mean phase observations for measuring the distance to M31 due to the small amplitudes of Cepheid light curves in the near-infrared (Madore & Freedman 1991). Because the Cepheid observations are obtained with the same near-infrared photometric system as recent distance scale data (Riess et al. 2011), the PHAT observations provide the means to determine the distance to M31 with lower systematic error than past estimates. The recent discovery and characterization of detached eclipsing binary (DEB) systems in M31 (Ribas et al. 2005; Vilardell et al. 2010) offer reliable and independent distance determinations to M31 which can be used to calibrate Cepheid luminosities and ultimately the Hubble constant. These measurements are also valuable for characterizing the slope of the period–luminosity relation at solar metallicity at long periods in the same near-infrared band used to determine the Hubble constant (Riess et al. 2011).

We have analyzed the first year of PHAT data to locate and measure the long-period Cepheids (log P(days) > 1) previously discovered by image subtraction in ground-based variability surveys, mostly from POMME. In Section 2, we present images of the recovered long-period Cepheids from the PHAT Survey and their near-infrared photometry. In Section 3, we analyze these data to constrain the distance to M31, the slope of the P–L relation in F160W, and its impact on the Hubble constant.

The PHAT Program (J. Dalcanton et al. 2012, to be submitted) is imaging the northeast quadrant of M31 with WFC3 with two UV filters (F275W, F336W), two IR filters (F110W, F160W), and two optical filters with ACS (F475W, F814W) over the course of a three-year survey. All WFC3 images are collected in a single epoch with those from ACS obtained approximately 6 months before or after the WFC3 images when the orientation of HST allows overlapping coverage in parallel. We searched the PHAT data for Cepheids with log P > 1 as these match the period range of Cepheids observed by HST at >20 Mpc used in studies of the distance scale. Our primary source of positions and periods of Cepheids was the recent POMME Survey from Megacam on CFHT whose sample of >2500 Cepheids covering most of M31 is the largest sample collected to date (J. Fliri et al. 2012, to be submitted), with 180 in the north half of M31 with log P > 1. While many of the Cepheids found in previous surveys are included in the POMME sample, the reverse is not true. We also consulted the variability catalog from the DIRECT Program (Bonanos et al. 2003) to search for additional long-period Cepheids not included in the POMME sample. In the first year PHAT obtained data through the middle of 2011, and 65 long-period POMME Cepheids were contained in the Survey, a fair fraction of the ∼90 likely to be imaged by the end of the Survey. These 65 were augmented with one object from DIRECT and two from the Pan-STARRS Survey data (S. Seitz et al. 2012, to be submitted).

Precise positions of the Cepheids in the HST images of M31 were determined by refining steps in the relative astrometry between ground- and space-based imaging. The ground-based survey positions were first used to identify the approximate region hosting the Cepheids in the HST images to ∼04 precision, with the uncertainty resulting from the HST guide star position errors. Next, a geometric transformation was defined by matching unresolved sources in 20'' diameter regions in the ground-based i-band POMME data and the WFC3 F110W images. The derived transformations were used with the POMME Cepheid coordinate to locate the Cepheid positions to within one WFC3-IR pixel (σ ∼ 01). For a few Cepheids with close neighbors, identifications of the Cepheids in the F336W images were used to confirm the Cepheid position among neighboring red giants (missing in the UV) or luminous blue dwarves (missing in the IR). Lastly, centroids for the Cepheids in the HST images were measured together with photometry resulting in the positions given in Table 1. The high signal-to-noise ratio of the Cepheid data in the HST images (S/N > 100) and minimal crowding (see Figure 1) ensure negligible bias in the measured positions.

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The F160W images for each pointing, consisting of 1600 s in four dithered exposures, were combined and resampled to a final scale of 008 pixel⁻¹. The Cepheid photometry was measured by simultaneously fitting model point-spread functions (PSFs) to the Cepheid and any unresolved sources in its vicinity using the same zero-point scale derived from the standard star P330E as in Riess et al. (2011). Artificial stars were added to the images and measured to assess the crowding bias and to determine the photometric errors. The mean bias was 0.002 mag and the mean statistical error was 0.01 mag. With the F110W filter, only a single dither of 700 or 800 s was obtained limiting the value of resampling the image on a finer scale and the use of PSF fitting, so we measured the F110W flux in small apertures of 2 pixel radius. Cepheid parameters are given in Table 1.

The F160W and F110W P–L relations are shown in Figure 2 with relevant parameters in Table 2. For historical interest, we have included one additional measurement in Figure 2, the HST WFC3-IR F110W observation of Hubble's (1929) first Cepheid discovered in M31, V1 with P = 31.4 days, observed by his namesake telescope by the Hubble Treasury Program Team (PI: K. Noll). The fitted slope of −3.00 ± 0.13 for F160W is in good agreement with the value of −2.91 ± 0.06 found for 448 more distant Cepheids in nine hosts with the same filter and instrument (Riess et al. 2011). The M31 slope is about 1.5σ shallower than the slope of −3.20 ± 0.06 we estimate for the LMC Cepheids from Persson et al. (2004) after interpolating between the ground-based J- and H-band slopes. The dispersion of the M31 Cepheids about this relation, 0.17 mag, is a factor of 3.5 times lower than that of the optical P–L relations measured for M31 from the ground and even less than the 0.23 mag dispersion measured by Macri et al. (2001) with NICMOS, likely a result of the greater photometric stability of WFC3-IR. Although the dispersion is larger than the 0.13 mag dispersion measured by Persson et al. (2004) in the H band in the LMC, the difference is readily explained by our use of random phase measurements. Resampling random phases from the Persson et al. (2004) light curves yields an average dispersion of 0.18 mag with no offset, a result nearly identical to ours, and confirming a similar intrinsic dispersion of 0.12 mag. A slope-insensitive distance indicator for our sample, the mean Cepheid magnitude at the sample mean period of log P = 1.2, is 18.292 ± 0.021 mag, sufficient to measure the distance to M31 to 1% given sufficient calibration.

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The dispersion increases to 0.20 mag for F110W. While some of the increase may result from additional differential extinction, most appears to come from the larger amplitudes of the light curves at shorter wavelengths. The LMC Cepheids predict a random phase dispersion of 0.22 mag. A strong correlation between the IR band residuals is apparent in Figure 2, as expected from our use of random but coincident phases as well as from intrinsic variation.

Next we fit a Wesenheit relation (PLW in Table 2) to account for the effect of differential extinction along the inclined line of sight of the form F160W − 1.54(F110W − F160W) = a + blog P where 1.54 is the value of A_F160W per magnitude of A_F110W − A_F160W for a Cardelli et al. (1989) reddening law with R_V = 3.1. The slope of this fit, −3.43 ± 0.17, is in good agreement with the slope of −3.38 ± 0.09 for the same relation in J and H for the LMC (Persson et al. 2004). A similar result was found using the F336W UV color in place of F110W with a reddening term of 0.14. Next we included a metallicity parameter (PLCZ in Table 2) to account for any apparent correlation between Cepheid near-infrared fluxes and the local value of 12 + log [O/H] measured from H ii regions, a proxy for Cepheid metallicity (see discussions of differing metallicity measurements in Kennicutt et al. 1998, Sakai et al. 2004, Bono et al. 2008, Shappee & Stanek 2011, Bresolin 2011, and Gerke et al. 2011). Use of the deprojected radial gradient from Zaritsky et al. (1994) results in an insignificant correlation of −0.65 ± 0.73 mag dex⁻¹, compared to −0.10 ± 0.09 mag dex⁻¹ from the extragalactic Cepheids in Riess et al. (2011). The M31 Cepheids have little grasp on the Cepheid metallicity parameter because they lie in a narrow annulus along the disk (J. Fliri et al. 2012, to be submitted) with a full range of less than 0.2 dex (12 + log [O/H] = 8.87–9.05).

To determine the distance to M31 as well as other parameters of interest we now make use of the sample of Cepheids measured in the near-infrared by Riess et al. (2011), the Milky Way parallax anchor for the Cepheid distance scale from Benedict et al. (2007), and SN data which can be used to determine the Hubble constant. We follow the same formalism as in Riess et al. (2011), solving one simultaneous system of linear equations which relate Cepheid magnitudes to their absolute magnitudes, slope of their P–L relations, local metallicity, distances to their hosts, and—with the inclusion of SN Ia data—a measurement of the Hubble constant. The Cepheids of M31 are assumed to share the same nuisance parameters as other Cepheids (i.e., luminosity, slope of P–L, metallicity dependence) but with a unique distance. For the M31 Cepheids, the photometric system used to measure their colors was somewhat different. While the Cepheids in the eight SN Ia hosts and M31 were all measured with F160W on WFC3-IR, the optical colors of the POMME M31 Cepheids, useful for dereddening, were not measured by Megacam or PHAT with the same V and I bands on HST as the others. To account for this difference we employed one of two different prescriptions: (1) we assumed a uniform value for the V − I color of the M31 Cepheids as the mean of those measured from the ground by the DIRECT Program, V − I = 1.23 ± 0.03 (indicated as fit PLW = H_{V, I} in Table 3) or (2) we used the individual F110W − F160W colors measured for the M31 Cepheids from the PHAT data with a small offset derived to give the same mean color correction in V − I from DIRECT⁴ (indicated as PLW = H_{J, H, X} in Table 3). The advantage of the latter approach is that it can account for differential reddening along the line of sight while providing a reddening correction which is consistent with that used for non-M31 Cepheids. We adopt a 0.03 mag systematic uncertainty for the use of colors measured with a different photometric system and a 0.04 mag systematic uncertainty between near-IR magnitudes of Cepheids measured on the ground and those measured from space.

Our best distance estimate for M31 is μ₀ = 24.38 ± 0.064, a 3.2% (statistical) distance determination which makes use of independent distance determinations to Milky Way Cepheid parallaxes.

The best-fit parameters are given in Table 3, with the preferred fit on the top line. Column 2 gives the value of χ²_dof, Column 3 the distance modulus of M31 and its statistical uncertainty, Column 4 the number of Cepheids used in the fit, Column 5 the value and total uncertainty in H₀ in km s⁻¹ Mpc⁻¹, Column 6 is a flag to indicate the use of Cepheids below the optical completeness limit (see below), Column 7 gives the metallicity calibration used, Column 8 the correlation coefficient in the PLWZ regression, and Column 9 the value and uncertainty of the slope of the Cepheid PLWZ relation. The next three columns are used to indicate variants in the analysis whose impact we now consider.

Ignoring the metallicity parameter (lines 2 and 4), changing the method of color correction (line 3), changing the reddening parameter for the Cepheids from R_V = 3.1 to R_V = 2.5 (line 6), or changing the metallicity scale from Zaritsky et al. (1994) to that of Bresolin (2011) (line 8) changes the distance to M31 by <0.02 mag. Failing to clip outliers in any Cepheid P–L relations (line 7) doubles the χ²_dof but changes the distance to M31 by <0.01 mag. Retaining infrared measurements of Cepheids with periods below the optically determined completeness limit (indicated by <P = N, line 8) has no effect on the distance. The only significant change in the distance to M31 occurs when discarding the color measurements used to account for extinction (line 5). The 0.16 mag increase in the distance in this case indicates that the M31 Cepheids have more extinction, ΔA_H = 0.16 mag (or ΔA_V = 0.75 mag) than the average of the Milky Way Cepheids. This is not surprising as M31 is highly inclined. It is possible to use additional anchors for the distance scale as in Riess et al. (2011) which can reduce the uncertainty in the distance to M31 below 3%.

Following the approach of Riess et al. (2011), we quantify the systematic uncertainty in the distance to M31 from the dispersion of the variants in the analysis, σ = 0.03 mag for the 12 variants not including the one neglecting color information which ignores the large extinction for the M31 Cepheids due to inclination. Thus our best estimate of the distance to M31 is μ₀ = 24.38 ± 0.064 ± 0.03 or 752 ± 27 kpc. This distance is in excellent agreement with the frequently cited measurement from Freedman & Madore (1990) and sits near the middle of the range of past measurements summarized by McConnachie et al. (2005).

Lastly, we made use of the two DEB measurements for M31 from Ribas et al. (2005) and Vilardell et al. (2010) with a mean of μ₀ = 24.36 ± 0.08 to determine the Hubble constant by their ability to calibrate the Cepheid luminosities. The present limitations of the use of M31 as an independent anchor are the lack of V − I colors measured with HST WFC3 and the lower precision of its independent distance compared to the other anchors (Riess et al. 2011). The use of M31 DEB results without any other anchors (line 10) yields a larger Hubble constant with larger error though still consistent with prior results. The best fit to the Hubble constant (line 1), which only makes use of the M31 Cepheids to constrain the slope and metallicity parameter, is 75.4 ± 2.9 km s⁻¹ Mpc⁻¹ and does not bring a significant difference to the one inferred by Riess et al. (2011) from the use of the Milky Way parallaxes.

The main advantages of the PHAT space-based observations of Cepheids presented here over previous data come from the reduction in extinction and crowding. The result is the tightest P–L relation for M31 Cepheids yet seen, which, combined with past, external calibrations, yields the most precise distance measurement for M31. The improved precision over the past history of optical, ground-based P–L relations is striking and continues to indicate the value of this kind of data for measuring the Hubble constant and dark energy (Riess et al. 2011).

To see more clearly the advantages of these spaced-based observations over those from the ground we simulated the effect of crowding for near-infrared Cepheid magnitudes obtained with good ground-based seeing by measuring the flux contained in an aperture of radius r = 09. The dispersion increased from 0.20 to 0.24 mag in F110W, the slope of the P–L relation flattened by 0.5, and the intercept became brighter by 0.3 mag. This is not unexpected as the impact of crowding is greater for the lower-period, fainter Cepheids. These results are similar to those determined for ground-based crowding in the optical by Mochejska et al. (2000). While crowding in ground-based observations is more severe in M31 than its distance would suggest due to its large inclination, the relative crowding can be even greater for HST observations of Cepheids in hosts of SNe Ia (Riess et al. 2009, 2011). In such cases artificial star measurements have been used to remove a photometric bias statistically, but cannot recover the precision of individual Cepheids. In addition, Cepheids found from the ground via image subtraction as in the POMME Survey are likely to suffer significantly greater crowding than those selected from PSF fitting as the addition of comparable constant flux to a PSF fit will reduce the amplitude of the light curve and remove the object from an amplitude-selected sample (Ferrarese et al. 2000).

The P–L relations for M31 may still improve in the near future. Additional observations by the PHAT Program should augment the Cepheid sample by dozens. Cepheids with log P < 1, though less useful for distance scale work, can be mined from the data to study the short-period end of the relation. The use of Cepheid phase information from concurrent, ground-based optical monitoring of M31 as demonstrated by Gerke et al. (2011) can be used to recover the phase of the PHAT observations, reducing the scatter of the P–L by up to ∼50%, an equivalent leverage as doubling the sample of measurements presented here.

We are grateful to the members of the PHAT MCT Program led by Julianne Dalcanton and aided by Jason Kalirai for their tremendous efforts to obtain the PHAT measurements and their support of this work. Financial support for this work was provided in part by the POMMME project (ANR 09-BLAN-0228). This work is also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the CFHT which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.

Facilities: HST - Hubble Space Telescope satellite, CFHT - Canada-France-Hawaii Telescope