THE CARNEGIE HUBBLE PROGRAM: THE LEAVITT LAW AT 3.6 AND 4.5 μm IN THE MILKY WAY

The 37 Cepheids in our sample have multiple distance and extinction estimates, and span a wide range of 4–70 days in period (Fernie et al. 1995; Tammann et al. 2003; Fouqué et al. 2007). The sample includes the majority of the nearest Cepheids; these should soon have high-precision parallaxes available from the Gaia mission (Windmark et al. 2011). Table 1 lists the target star, the adopted reddening, and three measurements of distance for each star if available: direct parallax data from HST, distance moduli obtained from main-sequence fitting of their host clusters, and distances from Infrared Surface Brightness (IRSB) measurements; the latter two methods were converted to units of parallax for comparison. Reddening estimates from photometric and spectroscopic methods are listed in Table 2 as well as space reddenings determined from stars along the same line of sight. The average reddening of these methods was used in Table 1.

Observations were made using the Spitzer Space Telescope as part of a two-year Exploration Science Program, PID 60010: The Hubble Constant (Freedman et al. 2008). The warm Spitzer mission started in 2009 (Cycle-6) and the Galactic Cepheid observations were completed in early 2011. Each Cepheid was observed at 24 epochs, pre-selected and scheduled to fully sample the light curve (23 epochs for ζ Gem). At each epoch, a nine-point dither pattern was used to mitigate array-dependent artifacts such as bad pixels and cosmic rays.

The majority of the data were taken using the sub-array mode of the IRAC³ (Fazio et al. 2004), with the shortest available frame time of 0.02 s (effective exposure time of 0.01 s). The sub-array mode outputs data from only one corner of the detector, in a 32 × 32 pixel format, thus allowing the shortest possible exposure times. The observations of CF Cas were made using the full-array mode (0.4 s frame time, 0.2 s effective exposure time) because it is relatively faint compared to the other program stars.

The sub-array data are provided by the Spitzer Science Center (SSC) in two forms: as an image cube of 64 frames (each 32 × 32 pixels) and as a single combined image; all further discussions to sub-array data refer to the combined form (sub2d extensions). All data were retrieved in the basic calibrated data (BCD) format, and were reduced using the most recent pipeline (S18.18.0).

Periods from the General Catalog of Variable Stars (Samus et al. 2009) were assumed. In the cases of U Car and V340 Nor, periods were computed using photometric data from Laney & Stobie (1992). Figure 1 presents the individual light and color curves (Vega magnitudes) for each Galactic Cepheid. Data points from Marengo et al. (2010) are shown as open triangles for comparison, when available. All are in good agreement except for U Car where the difference is likely due to a phase shift resulting from a period increase between the Laney & Stobie (1992) data and ours. All the light curves are plotted with the same magnitude range to emphasize relative changes in signal-to-noise ratio and amplitudes. The internal photometric precision is high, ranging from 0.004 to 0.029 mag. Interestingly, for the longer-period Cepheids, there is strong variation in the [3.6]–[4.5] color. This is due to the temperature-dependent carbon monoxide (CO) band in the [4.5] Channel, as will be discussed in Section 7.

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Smooth light curves were generated using a Gaussian local estimation (GLOESS) algorithm. GLOESS is an interpolating method that uses second-order polynomials to fit the data locally throughout the cycle. The data points surrounding the point to be fit are assigned weights according to a Gaussian window function; weights depend on their distance (in phase) from the fit point. The method has been used to conveniently obtain mean magnitudes by Persson et al. (2004) and Scowcroft et al. (2011) for LMC Cepheids, and by Monson & Pierce (2011) for Galactic Cepheids. These data are uniformly sampled so the error on the mean is $\sigma = {A}/{N\sqrt{12}}$, where A is the peak-to-peak amplitude of the light curve and N is the number of sample points; this is discussed in the Appendix of Scowcroft et al. (2011). The final total uncertainty in the mean magnitude turns out to be dominated by the systematic zero-point calibration of the Spitzer warm mission. Because the uncertainties in Channels 1 and 2 are correlated with each other, uncertainties in the color were also determined using the above equation. Table 4 gives the [3.6] and [4.5] IRAC intensity-mean magnitudes and colors for the 37 Cepheids.

Before discussing the period–luminosity and period–color relationships, we shall need to correct for extinction. Compared to optical wavelengths, reddening, and extinction corrections are relatively small at mid-IR wavelengths. They must, nevertheless, be quantified and applied because values of A_V can exceed 3 mag in our sample. We adopted an extinction law for all stars that is applicable along average lines of sight through the diffuse interstellar medium. The extinction law of Indebetouw et al. (2005) combined with that of Cardelli et al. (1989) best fulfill this choice. The relations A_[3.6]/A_K = 0.56 ± 0.06 and A_[4.5]/A_K = 0.43 ± 0.08 (Indebetouw et al. 2005) were derived from field stars in the Galactic plane and are probably applicable to the Cepheids in this study.⁷

To scale the extinctions at K to the reddenings E(B − V) we used the extinction law derived by Cardelli et al. (1989): A(λ)/A_V = a(x) + b(x)/R_V, where a = 0.574x^1.61, b = −0.527x^1.61, x = 1/λ, and R_V is the ratio of total-to-selective absorption (R_V = A_V/E(B − V)). Using a wavelength of λ = 2.164 μm for the K filter (actually K_s) as adopted by Indebetouw et al. (2005) and an average R_V = 3.1, we find A_K/A_V = 0.117. The combined relations yield a final total-to-selective extinction of A_[3.6]/E[B − V] = 0.203, A_[4.5]/E[B − V] = 0.156, and E([3.6] − [4.5])/E[B − V] = 0.047.

Use of the Indebetouw et al. (2005) mid-IR extinction law might be questioned on general grounds. Three stars have E(B − V) > 1, above which the corrections will begin to introduce systematic errors. For example, measured values of A_[3.6]/A_K range from 0.41(Chapman et al. 2009) to 0.64 (Flaherty et al. 2007).⁸ Toward the Galactic center Nishiyama et al. (2009) obtain 0.50 ± 0.01. For V367 Sct (E(B − V) = 1.231) the total range in A_[3.6] is 0.10 mag. The uncertainties in E(B − V) are all ⩽0.03 mag except for GY Sge, where σ(E(B − V)) = 0.17. This value introduces uncertainties of 0.035 and 0.027 mag into the 3.6 and 4.5 μm magnitudes, respectively. Finally, if the uncertainty in A_[3.6]/E[B − V] is as large as 0.05 (see above), the corresponding corrections will remain negligibly small.

Of the 37 Cepheids, 10 have direct geometric parallaxes determined from HST guide camera data (Benedict et al. 2007) and therefore have the most accurate distance determinations currently available. The data for the sample are provided in Table 5. Figure 2 shows the data and zero-point fit using uniform weighting; the distance uncertainties are displayed for reference as error bars. The data include the final de-reddened [3.6] and [4.5] mag, the final adopted distance moduli, extinctions, and absolute magnitudes. Following Benedict et al. (2007) we have applied Lutz–Kelker–Hanson corrections (Lutz & Kelker 1973; Hanson 1979, LKH) to these parallaxes; the corrections are systematic and range from −0.02 to −0.15 mag with uncertainties of ±0.01 mag. For completeness we include in Table 6 and Figure 3 the corresponding data for the HST sample without LKH corrections.

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Of the 37 Cepheids, 18 are likely to be members of star clusters or associations for which distance moduli have been estimated from main-sequence (MS) fitting (Turner & Burke 2002; Turner 2010; Majaess et al. 2011, 2012a, 2012b). Table 7 lists the data and Figure 4 shows the forced LMC slope and zero-point fit for the uniformly weighted data. The Cepheids CEa and CEb Cas are presumably at the same distance as CF Cas by virtue of common membership in the cluster NGC 7790 and although separated by only 10 they were easily split using point response function (PRF) photometry and they have been included in the sample.

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Of the 37 Cepheids, 32 have distance determinations based on the IRSB technique (Storm et al. 2011b, and references therein). Table 8 contains the data and Figure 5 shows the fit and zero point for uniformly weighted data; W Sgr was rejected from the analysis due to its relatively high uncertainty.

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Each of the data subsamples were fit using a PL relationship of the form: M = a(log P − 1) + b. As will be shown below we find no statistically significant difference between the slope of the [3.6] PL of −3.31 ± 0.05 for the LMC and that of the HST parallax sample. We thus adopt the LMC slope for the PL fit and re-determine zero points for each subsample. The magnitude residuals from the PL fits are highly correlated with each other, suggesting that the widths of the PL relations are not driven by random photometric errors. Rather, the correlated scatter is most likely some combination of deterministically correlated (unit slope) distance errors and the intrinsic (correlated) positions of these Cepheids in the instability strip (IS). If the IS is represented by a rectangular distribution (i.e., it is uniformly filled and has hard limits at the blue and red edges) then the peak-to-peak width in the residuals can be interpreted as the width of the IS or at least an upper limit, which in the HST subsample is ∼0.4 mag.

The PL relations were fit to each of the three subsamples using multiple weighting schemes and also by further restricting the subsamples by period cuts.

The final uncertainty in absolute magnitude for an individual Cepheid is dominated by the uncertainty in its distance. In deriving a PL relation, however, an additional spread is caused by the finite width of the IS, and biases in the PL slope and zero point may result depending on how the strip is filled. To investigate these uncertainties and their effects on the derived PL fits, we applied different weighting schemes. In addition, for each of the data sets, we investigated different period cuts so that the Galaxy data sets more closely matched the period range of the LMC sample, viz., 6–60 days. Finally, a fixed slope determined from the LMC data was force fitted to the Galactic data to determine only the Galactic zero point. The data were fit using a PL relationship of the form M = a(log P − 1)  +  b. Table 9 presents the results using three different weighting methods for the data in this analysis.

Table 9. Period–Luminosity (Leavitt) Laws

Galaxy	Sample	Band	N	weights= 1/0.12			weights = 1/σ2M			weights = 1/(σ2M + 0.12)
				(slope) a	(intercept) b	σ	(slope) a	(intercept) b	σ	(slope) a	(intercept) b	σ
MW	πHST	[3.6]	10	−3.40 ± 0.12	−5.81 ± 0.04	0.09	−3.35 ± 0.22	−5.80 ± 0.06	0.10	−3.39 ± 0.26	−5.80 ± 0.07	0.09
MW	πHST 6 ⩽ P ⩽ 60	[3.6]	5	−3.33 ± 0.18	−5.82 ± 0.05	0.09	−3.33 ± 0.35	−5.79 ± 0.06	0.08	−3.34 ± 0.39	−5.80 ± 0.08	0.08
MW	πHST no LKH	[3.6]	10	−3.40 ± 0.12	−5.75 ± 0.04	0.09	−3.33 ± 0.22	−5.76 ± 0.06	0.10	−3.37 ± 0.26	−5.76 ± 0.07	0.10
MW	πHST no LKH 6 ⩽ P ⩽ 60	[3.6]	5	−3.27 ± 0.18	−5.78 ± 0.05	0.07	−3.27 ± 0.35	−5.75 ± 0.06	0.06	−3.28 ± 0.39	−5.76 ± 0.08	0.07
MW	πMS	[3.6]	18	−3.00 ± 0.07	−5.76 ± 0.02	0.19	−3.43 ± 0.08	−5.77 ± 0.02	0.17	−3.11 ± 0.12	−5.75 ± 0.03	0.19
MW	πMS 6 ⩽ P ⩽ 60	[3.6]	11	−3.30 ± 0.12	−5.68 ± 0.03	0.19	−3.70 ± 0.10	−5.74 ± 0.02	0.17	−3.37 ± 0.17	−5.70 ± 0.04	0.19
MW	πIRSB	[3.6]	34	−3.42 ± 0.05	−5.73 ± 0.02	0.13	−3.39 ± 0.02	−5.74 ± 0.01	0.13	−3.41 ± 0.06	−5.73 ± 0.02	0.13
MW	πIRSB 6 ⩽ P ⩽ 60	[3.6]	23	−3.35 ± 0.08	−5.75 ± 0.02	0.13	−3.31 ± 0.03	−5.77 ± 0.01	0.14	−3.33 ± 0.09	−5.75 ± 0.03	0.13
LMCa	6 ⩽ P ⩽ 60	[3.6]	80	$\bf {-3.31\pm 0.05 }$	$\bf {12.70 \pm 0.02}$	0.11	−3.37 ± 0.07	12.71 ± 0.02	0.11	−3.31 ± 0.05	12.70 ± 0.02	0.11
MWb	πHST	[3.6]	10	...	$\bf {-5.80 \pm 0.03}$	0.10	...	−5.79 ± 0.05	0.10	...	−5.79 ± 0.06	0.10
MWb	πHST 6 ⩽ P ⩽ 60	[3.6]	5	...	−5.82 ± 0.04	0.09	...	−5.79 ± 0.06	0.08	...	−5.80 ± 0.08	0.08
MWb	πHST no LKH	[3.6]	10	...	−5.74 ± 0.03	0.10	...	−5.76 ± 0.05	0.10	...	−5.75 ± 0.06	0.10
MWb	πHST no LKH 6 ⩽ P ⩽ 60	[3.6]	5	...	−5.77 ± 0.04	0.07	...	−5.75 ± 0.06	0.06	...	−5.76 ± 0.08	0.07
MWb	πMS	[3.6]	18	...	−5.75 ± 0.02	0.21	...	−5.76 ± 0.02	0.17	...	−5.76 ± 0.03	0.19
MWb	πMS 6 ⩽ P ⩽ 60	[3.6]	11	...	−5.67 ± 0.03	0.19	...	−5.74 ± 0.02	0.18	...	−5.71 ± 0.04	0.19
MWb	πIRSB	[3.6]	34	...	−5.74 ± 0.02	0.14	...	−5.75 ± 0.01	0.13	...	−5.74 ± 0.02	0.13
MWb	πIRSB 6 ⩽ P ⩽ 60	[3.6]	23	...	−5.75 ± 0.02	0.13	...	−5.77 ± 0.01	0.14	...	−5.75 ± 0.02	0.13
MW	πHST	[4.5]	10	−3.26 ± 0.12	−5.78 ± 0.04	0.09	−3.23 ± 0.22	−5.77 ± 0.06	0.09	−3.26 ± 0.26	−5.77 ± 0.07	0.09
MW	πHST 6 ⩽ P ⩽ 60	[4.5]	5	−3.13 ± 0.18	−5.80 ± 0.05	0.08	−3.15 ± 0.35	−5.77 ± 0.06	0.08	−3.15 ± 0.39	−5.77 ± 0.08	0.08
MW	πHST no LKH	[4.5]	10	−3.27 ± 0.12	−5.72 ± 0.04	0.09	−3.21 ± 0.22	−5.74 ± 0.06	0.10	−3.25 ± 0.26	−5.73 ± 0.07	0.10
MW	πHST no LKH 6 ⩽ P ⩽ 60	[4.5]	5	−3.07 ± 0.18	−5.76 ± 0.05	0.07	−3.09 ± 0.35	−5.73 ± 0.06	0.06	−3.09 ± 0.39	−5.74 ± 0.08	0.06
MW	πMS	[4.5]	18	−2.91 ± 0.07	−5.72 ± 0.02	0.19	−3.33 ± 0.08	−5.73 ± 0.02	0.17	−3.01 ± 0.12	−5.71 ± 0.03	0.19
MW	πMS 6 ⩽ P ⩽ 60	[4.5]	11	−3.19 ± 0.12	−5.64 ± 0.03	0.19	−3.61 ± 0.10	−5.71 ± 0.02	0.17	−3.27 ± 0.17	−5.66 ± 0.04	0.20
MW	πIRSB	[4.5]	34	−3.32 ± 0.05	−5.70 ± 0.02	0.13	−3.28 ± 0.02	−5.71 ± 0.01	0.13	−3.31 ± 0.06	−5.70 ± 0.02	0.13
MW	πIRSB 6 ⩽ P ⩽ 60	[4.5]	23	−3.23 ± 0.08	−5.72 ± 0.02	0.12	−3.18 ± 0.03	−5.74 ± 0.01	0.12	−3.22 ± 0.09	−5.72 ± 0.03	0.12
LMCa	6 ⩽ P ⩽ 60	[4.5]	80	−3.21 ± 0.06	12.69 ± 0.02	0.12	−3.28 ± 0.08	12.71 ± 0.03	0.12	−3.21 ± 0.06	12.70 ± 0.02	0.12
MWb	πHST	[4.5]	10	...	−5.77 ± 0.03	0.10	...	−5.77 ± 0.05	0.09	...	−5.76 ± 0.06	0.10
MWb	πHST 6 ⩽ P ⩽ 60	[4.5]	5	...	−5.79 ± 0.04	0.09	...	−5.77 ± 0.06	0.08	...	−5.77 ± 0.08	0.08
MWb	πHST no LKH	[4.5]	10	...	−5.71 ± 0.03	0.10	...	−5.73 ± 0.05	0.10	...	−5.73 ± 0.06	0.10
MWb	πHST no LKH 6 ⩽ P ⩽ 60	[4.5]	5	...	−5.75 ± 0.04	0.08	...	−5.73 ± 0.06	0.07	...	−5.74 ± 0.08	0.07
MWb	πMS	[4.5]	18	...	−5.72 ± 0.02	0.21	...	−5.72 ± 0.02	0.18	...	−5.72 ± 0.03	0.20
MWb	πMS 6 ⩽ P ⩽ 60	[4.5]	11	...	−5.63 ± 0.03	0.19	...	−5.70 ± 0.02	0.18	...	−5.67 ± 0.04	0.20
MWb	πIRSB	[4.5]	34	...	−5.71 ± 0.02	0.13	...	−5.72 ± 0.01	0.13	...	−5.71 ± 0.02	0.13
MWb	πIRSB 6 ⩽ P ⩽ 60	[4.5]	23	...	−5.72 ± 0.02	0.12	...	−5.73 ± 0.01	0.13	...	−5.72 ± 0.02	0.12

Notes. The form of the PL relation used in these fits is M = a(log P − 1) + b. The values in bold indicate our adopted PL slope and zero points. To eliminate asymmetric rounding errors when reporting two significant figures in the zero points a floor rounding function was used which rounds toward negative infinity rather than away from zero. ^aThe LMC sample does not contain Cepheids with periods less than six days. The LMC data are discussed in Scowcroft et al. (2011). ^bForce fit the LMC slope from the unweighting method to find the zero point.

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The first weighting method applies a uniform uncertainty of 0.1 mag to each Cepheid, the purpose of which is to provide results presumably less biased by Cepheids which may have underestimated distance uncertainties. The second method applies traditional weights as σ⁻² to the absolute magnitudes. The third method falls between the first two in that it assumes an intrinsic scatter in the IS. In this case, an additional uncertainty of 0.1 mag is added in quadrature to each individual uncertainty. The value of 0.1 mag is adopted from the average rms scatter of the LMC data points around the best fit.

The IRSB slopes closely match the HST slopes, which is not surprising because the most recent IRSB distances used a projection factor (p-factor) calibrated using the HST parallaxes (but without LKH corrections) as priors. The IRSB slope is better constrained because of sample size, but is still dependent on the adopted p-factor. The effect of varying the weighting method is most noticeable in the Cluster MS fits where the slopes differ by more than 2σ between the first two methods, and converges to within 1σ of the HST, IRSB, and LMC slopes using the third weighting method. As discussed by Turner (2010) some long-period Cepheids populate the blue edge of the IS and can bias the slope, so it is necessary to include an estimate of the intrinsic width of the IS to reduce the bias. The benefit of the third weighting method is that the intrinsic width of the IS is included as well as individual uncertainties for each Cepheid. As can be seen in Table 9 the slopes for all three methods agree very well with each other using the softened weights. Since the slope from each method agrees with that of the LMC, we chose to adopt the better-determined LMC slope and to redetermine the zero points for both the [3.6] and [4.5] PL relations. This decision is further backed by recent studies that find near-identical PL slopes for the MW and LMC in the near-infrared (Storm et al. 2011a).

As mentioned above, some of the long-period Cepheids occupy the blue edge of the IS and although we forced a fixed (LMC) slope, the zero point can now be slightly biased if the entire sample does not uniformly populate the IS. We therefore limit ourselves to adopt zero points from the uniformly weighted fits. This effectively assumes the width of the IS is the only source of uncertainty and can be treated as equal for each Cepheid. We also chose to make use of the entire period range which provides a larger sample and will more uniformly populate the IS. With these choices in hand, we now have zero points of HST (with LKH) = −5.80  ±  0.03, HST (without LKH) = −5.74  ±  0.03, MS = −5.75  ±  0.05, and IRSB = −5.74  ±  0.02. We note again the agreement between the HST (without LKH) and IRSB zero points as must be the case (see above). The average LKH correction is −0.06 mag, which if applied to the IRSB calibration would shift the [3.6] IRSB PL zero point to −5.80  ±  0.03 mag. Because they are calibrated using the HST parallaxes the IRSB zero point does not offer an independent baseline measurement; however, it does sample the IS better and since it yields the same zero point and scatter, it indicates that the HST data are not too affected by paucity.

The Cluster Cepheids do offer an independent check of the zero point and they appear to confirm the HST (without LKH) zero point. We note, however, that the outliers S Vul and TW Nor are more than 10% discrepant compared to their IRSB distances and rejecting them would change the Cluster zero point to −5.79 mag, in agreement with LKH. This is in agreement with the Ngeow (2012) results which also find a 0.04 mag relative offset between their derived Wesenheit PL distances to the Storm et al. (2011b) IRSB and Turner (2010) Cluster samples. Based on discussions in the literature on the use of LKH (Lutz & Kelker 1973; Hanson 1979; Smith 2003; Loredo 2007) we have, at present, chosen to adopt the use of the LKH correction and an uncertainty in this correction of 0.01 mag (Benedict et al. 2007).

We note that the scatter in the HST data is less than the average uncertainty assigned to the HST parallaxes and that we have chosen to adopt this (smaller) empirical scatter as a measure of the total zero-point uncertainty.

CO absorption in the 4.5 μm band for cooler stars produces a significant period–color relation that moves the mean [3.6]–[4.5] color toward the blue at cooler temperatures, and longer periods. This effect is driven, as is the case for the light curve color variations (see below), by the temperature dependence of CO dissociation and not by the thermal color–temperature which has little effect on the slope of the continuum at these long wavelengths. The [3.6]–[4.5] period–color relation is shown in Figure 8 and the weighted least-squares fit to the de-reddened data (omitting Y Oph) is [3.6]–[4.5]_MW = −0.09(± 0.01)(log P − 1.0) − 0.03(± 0.01). For comparison, the LMC period–color relation is [3.6]–[4.5]_LMC = −0.09(± 0.01)(log P − 1.0) + 0.01(± 0.01) (Scowcroft et al. 2011). The slopes of these fits are consistent, but the zero points differ by 0.04 ± 0.01 mag.

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Ngeow et al. (2012) have presented a number of models for theoretical PC diagrams in the mid-IR. Their summary tables give slopes and zero points for several models of the PLs and [3.6]–[4.5] color covering a range of helium and metal abundance. Several of their model PCs are plotted with the data in Figure 8. Comparison of the theoretical slope of the [3.6]–[4.5] PC diagram shows reasonably good agreement with the Galaxy color data for a helium abundance of Y = 0.31.

As pointed out in Section 4.1, the [3.6]–[4.5] color is characterized by systematic variations through the cycle. The color amplitude is also closely related to period, the longer-period stars having the largest variations. This effect, in complete analogy with the cause of the period–color relation, is again due to CO absorption in the 4.5 μm band, as discussed by Scowcroft et al. (2011) and Marengo et al. (2010). The color variation extends only toward bluer colors from a baseline red color limit of ∼0.01 mag; this can be seen Figure 9. The blue extent (blue indicating more absorption at 4.5 μm) increases with period as the Cepheids reach intrinsically cooler temperatures. The effect of CO has only recently been observed over entire Cepheid pulsation cycles (see also Scowcroft et al. 2011).

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To quantify the behavior of both the overall PC relation and the color curves, we have computed several synthetic spectra using appropriate Kurucz stellar models (Kurucz 1993; Castelli & Kurucz 2003; Sbordone et al. 2004; Sbordone 2005). Figure 10 shows the results. They indicate that at temperatures greater than approximately 6000 K absorption due to CO is nearly non-existent. As the temperature falls below 6000 K CO absorption in the [4.5] band sets in, leading to the diminished flux observed in the 4.5 μm light curve. The result is that the color curves should have larger amplitudes for Cepheids with longer periods, as they reach intrinsically cooler temperatures. This is precisely the behavior exhibited in the observed color curves. For the shorter-period Cepheids the color amplitude is diminished because these Cepheids are intrinsically hotter and the CO remains dissociated over a longer portion of the pulsation cycle.

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Part of the systematic offset in the PC fits (Galaxy versus LMC) is plausibly explained by the difference in metal abundance between the two galaxies. Figure 10 shows that this shift should amount to ∼0.02 mag offset per 0.5 dex change in metal abundance. Other effects such as rotation may also play a role, and in any case we are dealing with a very small effect.

Saturated pixels. The first step in the reduction is to find and mask markedly nonlinear or saturated pixels. This is particularly important for stars as bright as the Cepheids in this program. Our routine works as follows: MOPEX fits the profile by masking unwanted pixels, and by assigning weights using the uncertainty image. Saturated pixels (near the center of the point-spread function (PSF)) will tend to have lower dispersions and hence artificially high weights (in the limit of complete saturation the dispersion will be zero). We found valid upper thresholds by experiment: the dispersions for the nine dither positions of a saturated star were found as a function of threshold level and the best level chosen. The final upper threshold values were 10,000 and 12,000 DN for the 3.6 and 4.5 μm bands, respectively. These were lower than those recommended in the IRAC Handbook.

Point response profiles and pixel phase corrections. Each stellar profile was fit with a PRF profile, a procedure that minimizes the residual between the input frame and a standard PRF supplied by MOPEX. Rather than having a functional form, the PRF is a look-up table containing different representations of a point source at various pixel phases. These are the distances from the center of the stellar profile to the center of the nearest (integer valued) pixel. This complication arises because pixels do not have uniform response across them. The pixel phase correction (PPC) was included in the PRF tables provided by the SSC for the cold mission. This was not the case for the warm mission. The correction for an arbitrary profile was thus determined using all the data to find a empirical PPC as follows. For every nine-point dither pattern constituting a measurement one has an average count, and nine deviations from that count. (The deviations arise from the pixel phase variation.) The left-hand side of Figure 11 shows all those deviations plotted as a function of pixel phase. The total number of points is 9 × 24 × 37 (9 dither positions per measurement, 24 light curve points per star, 37 stars). The strong correlation represents the residual PPC, which is easily removed to yield corrected data. The correction is of the form $f_{\rm {ppc}} = u+v (({1}/{\sqrt{2\pi }}) - p)$, where $p={\sqrt{((x-{\rm {nint}}(x))^2 + (y-{\rm {nint}}(y))^2)}}$ is the pixel phase.⁹ The right-hand side of Figure 11 shows the results of applying the above correction to the data points on the left-hand side. The results are seen to be satisfactory.

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The PRF photometry methods correct for several other systematic effects.

• 1.  
  The fits have slightly different coefficients for saturated and unsaturated data, a systematic effect that probably arises from centroid offsets in masking central pixels. The coefficients were confirmed by masking non-saturated data.
• 2.  
  The PRF varies across the array, but a fixed PRF was used for the sub-array data, located at column 233, row 35. For the full frame data (CF Cas) a lookup table was used to find the nearest PRF at each position.
• 3.  
  MOPEX reports fluxes at the center of pixel flux and is normalized to a radius of 10 pixels, i.e., the IRAC standard aperture. Because the flux reported is that for a pixel phase of zero, an additional correction factor (f_corr) is applied to bring the flux to the average pixel phase, which corresponds to where the flux zero points were defined.

Finally, the flux is placed on the standard Vega magnitude system by dividing by the photometric flux zero points (zp); 280.9 ± 4.1 and 179.7 ± 2.6 [Jy], for [3.6] and [4.5], respectively (Reach et al. 2005). The final-corrected magnitude (m) is found from the PRF flux (F_PRF), which is reported in μJy, by the following relation:

$$$ m = -2.5 \log \left(\frac{F_{\rm {PRF}}\cdot 10^{-6}}{f_{\rm {corr}}\cdot f_{\rm {ppc}}\cdot zp} \right). $$$

Table 12 contains the constants for each channel.

Image persistence or latency. The sub-array data were taken with short exposure times on bright objects, and with short settling times between dithers. As a consequence, the data are prone to short-term image persistence from previous dithers and observations; see Figure 12. A multi-stage process was undertaken to mitigate image persistence for each frame. First, the stellar profile was fit in each of the nine dithered frames using the PRF-fitting algorithm in the MOPEX script described above. The residual images were averaged together using a nearest neighbor weighting scheme for each dither position. Lowest weights were given to future frames and higher weights were given to the most recent frames to create an approximate map of the image persistence at each dither position. The persistence maps were then subtracted from the original data leaving only the source; see Figure 13. Note that this process simply describes creating a local background frame by combining dithered frames where the source(s) has been modeled and subtracted rather than masked. Each background/persistence-subtracted image was passed through the MOPEX pipeline to perform a second and final PRF fit.

The latency can affect aperture photometry measurements by nearly 5% in the short (0.02 s) sub-array data. The effect is larger for shorter exposure times when there is less time for the latency to dissipate. One advantage of the PRF-fitting algorithm is that it tends to ignore the latent pixels when fitting a PRF. After the latent image subtraction the PRF photometry changes by less than 1%.

Photometry checks. As a check of the photometric fidelity the standard star HD165459 was processed in the manner described above for various exposure times in sub-array and full-array mode resulting in non-saturated and saturated data. To quantify the effect of persistence both aperture and PRF photometry were performed on the standard star both prior to, and after, the persistence correction. Prior to the correction, the aperture photometry was consistently reporting 5%–15% higher flux than expected for the shortest exposure times while after the persistence correction the aperture photometry was typically only 0%–5% higher than expected. The PRF photometry was relatively unaffected by the persistence correction, changing by less than 1% before and after the persistence correction. The standard star comparisons represent the worse case scenario since the long exposures (saturated data) were taken just prior to the short exposures and subsequently suffered from a relatively large amount of persistence. The final PRF photometry results for the non-saturated standard data are 6.588 ± 0.007 mag and 6.571 ± 0.008 mag and 6.587 ± 0.014 mag and 6.564 ± 0.017 mag for the saturated data. Both are in good agreement with the standard magnitudes of 6.593 ± 0.029 mag and 6.575 ± 0.028 mag found by Reach et al. (2005). For the Cepheids, the difference in PRF photometry before and after the persistence correction was also less than 1%; except for ℓ Car, the most severely saturated star in our sample, in which case the difference was 6% and 3% in Channels 1 and 2. We report here the final PRF photometry resulting from a persistence-subtracted image.