THE ACS LCID PROJECT. III. THE STAR FORMATION HISTORY OF THE CETUS dSph GALAXY: A POST-REIONIZATION FOSSIL^*

After testing different approaches to drizzle the set of images, we opted to work on the original _FLT data. A custom dithering pattern was adopted for the observations. The shifts applied were small enough (<10 pixels) that the camera distortions would not cause problems when subsequently merging the catalogs from different images. The two chips of the camera have been treated independently, and we applied the following steps to all the individual images.

• 1.  
  Basic reduction. We adopted the basic reduction (bias, flat field) from the On-The-Fly-Reprocessing pipeline (CALACS v.4.6.1, 2006 March 13). Since we opted for the _FLT images, we applied the pixel-area mask to account for the different area of the sky projected onto each pixel, depending on its position.
• 2.  
  Cosmic ray rejection. We used the data quality files from the pipeline to mask the cosmic rays. According to the flag definition in the ACS Data handbook, pixels with the value 4096 are affected by cosmic rays. These pixels were flagged with a value higher than the saturation threshold. This allowed DAOPHOT to recognize and properly ignore them.
• 3.  
  Aperture photometry. We performed the source detection at a 3σ level and aperture photometry, with aperture radius = 3 pixels.
• 4.  
  PSF modeling. Careful selection of the PSF stars was done through an iterative procedure. Automated routines were used to identify bright, isolated stars, with good shape parameters (sharpness) and small photometric errors. Point-spread function (PSF) stars were selected in order to sample the whole area of the chip, to properly take into account the spatial variations of the PSF. The last step was a visual inspection of all the PSF stars, for all the images, to reject those with peculiarities like bright neighbors, contamination by cosmic rays, or damaged columns. On average, we ended up with more than 150 PSF stars per image.

We performed many tests in order to understand which PSF model, among those available in DAOPHOT, would work best with our data. We used the same sets of stars for each fixed analytical PSF model, and tested different degrees of variation across the field, from purely analytical and constant (variability(va) = −1, 0) to varying cubically (va =3). We evaluated the quality of the final CMD, and the best results were obtained with va >0, but the tightness of the CMD sequences did not vary appreciably when using a quadratically or cubically variable PSF. We also checked the photometric errors and sharpness estimators, but could not find any evident improvement when increasing the degree of variation of the PSF. We therefore adopted a simpler and faster linear PSF, allowing the software to choose the analytical function on the basis of the χ² estimated by the PSF routine. In all the images, a Moffat function with index β = 1.5 was selected.

The next step was ALLSTAR profile-fitting photometry for individual images. The resulting catalogs were used only to calculate the geometric transformations with respect to a common coordinate system, using DAOMATCH and DAOMASTER. Geometrical transformations are needed to generate a stacked image with MONTAGE2. The resulting median image, cleaned from cosmic rays, was used to create the input list for ALLFRAME. Rejecting all the objects with |sharp|>0.3 was an efficient way to remove most of the extended objects (i.e., background galaxies) as well as remaining cosmic rays. After the first ALLFRAME was run, we used the updated and more precise positions and magnitudes to refine the coordinate transformations and to recalculate the PSFs, using the previously selected list of stars. A second ALLFRAME was finally run with updated geometric transformations and PSFs.

The calibration to the VEGAMAG system was done following the prescriptions given by Sirianni et al. (2005), using the updated zero points from Mack et al. (2007; because Cetus was observed after the temperature change of the camera in 2006 July). Particular attention was paid to calculate the aperture correction to the default aperture radius of 05. Aperture photometry in various apertures was performed on the PSF stars of every image, once the neighboring stars had been subtracted. The growth curve and total magnitude of the stars were obtained using DAOGROW (Stetson 1990). An aperture correction was calculated for every image, with the result that the mean correction was 0.029 ± 0.009 mag. Individual catalogs were calibrated, and the final list of stars was obtained keeping all the objects present in at least (half + 1) images in both bands. The final CMD is presented and discussed in Section 3.

To estimate the completeness and errors affecting our photometry, we adopted a standard method of artificial-star tests. We added ∼350,000 stars per chip, adopting a regular grid where the stars were located at the vertices of equilateral triangles. The distance between two artificial stars was fixed in order to pack the largest number of stars but at the same time to ensure that the synthetic stars were not affecting each other. This value was chosen to be R = (rad_PSF + rad_fit + 1) = 13 pixels. In this way, overlapping of the wings of the artificial stars was minimized to avoid influencing the fit in the core of the PSF. With this criterion, taking into account the dimension of the CCD, we needed seven iterations with ∼49,770 stars each. In each iteration, the grid was moved by a few pixels, to better sample the crowding characteristics of the image.

To verify the hypothesis that the synthetic stars were not affecting each other, we also performed completeness tests injecting stars separated by (2 × rad_PSF + 1) = 21 pixels. We did not find any significant difference in the completeness level of synthetic stars, meaning that the original separation was adequate. With this criterion, the incompleteness of the synthetic stars only depends on the real stars, while the additional crowding due to the partial overlap of the injected stars is negligible.

The list of injected stars was selected following the prescriptions introduced by Gallart et al. (1996). To fully sample the observed magnitude and color ranges, we adopted a synthetic CMD created with IAC-star (Aparicio & Gallart 2004), adopting a constant star formation rate (SFR) between 0 and 15 Gyr, with the metallicity uniformly distributed between Z = 0.0001 and Z = 0.005 at any age. The stars were added on the individual images, taking into account the coordinate and magnitude shifts. The photometry was then repeated with the same prescriptions as for the original images. The results are summarized in Figure 3, where the Δmag = mag_in − mag_out of the artificial stars and selected completeness levels are shown for the two bands, as a function of the input magnitude.

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DOLPHOT is a general-purpose photometry package adapted from the WFPC2-specific package HSTphot (Dolphin 2000b).¹⁶ For this reduction, we used the ACS module of DOLPHOT and followed the recommended photometry recipe provided in the manual for version 1.0.3. The procedure is fairly automated, with the software locating stars, adjusting the default Tiny Tim PSFs based on the image, iterating until the photometry converges, and making aperture corrections. Similar to the procedures with DAOPHOT, we used the original _FLT images, corrected for the pixel-area mask and cosmic rays. We used a drizzled image from the HST archive as the reference for the coordinate transformations. The search for stellar sources was performed with a threshold t = 2.5σ. The photometry was done fitting a model PSF calculated with TinyTim. DOLPHOT also automatically estimated the aperture correction to 05. The same calibration as for the DAOPHOT photometry was used.

Artificial-star tests were performed by injecting ∼140, 000 synthetic stars in the original images. The input colors and magnitudes of the synthetic stars cover the complete range of the observed colors and magnitudes. The stars uniformly sample the range −1 mag < M_F475W − M_814W < 5 mag, −7 mag < M_F475W < 8 mag. The photometry was performed following the procedure suggested by Holtzman et al. (2006). The synthetic stars were injected one at a time into the images to avoid affecting the stellar crowding in the images. Stars were uniformly distributed to cover the whole area of the camera. Figure 4 shows Δmag = mag_in− mag_out of the artificial stars together with the 50% and 90% completeness levels for the two bands.

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A direct comparison between the DAOPHOT and DOLPHOT photometry is presented in Figure 5. We detect a small zero-point offset for the brightest stars in both bands ((m_F475W,DAO − m_F475W,DOL) ∼ 0.023 mag, for m_F475W< 25, and (m_F814W,DAO − m_F814W,DOL) ∼ −0.010 mag, for m_F814W< 24), and a trend as a function of magnitude: moving toward fainter magnitudes, DAOPHOT tends to give slightly fainter results (up to Δm_F475W≃ 0.05 mag, at m_F475W = 29.1, Δm_F814W ≃ 0.07 at m_F814W = 28.0). Note that since the trend with the magnitude is similar in the two bands, the color difference shows a residual zero-point offset (∼0.04 mag), but no dependence on magnitude. We performed an exhaustive search for the possible cause of such small differences, but its origin remained elusive, and it is likely a result of the many differences between the approaches of the two photometric packages (PSF, sky determination, aperture correction calculation). Note that similar trends were detected and discussed by Holtzman et al. (2006) and Hill et al. (1998) when comparing different photometric codes applied to HST data. This reinforces the idea that small differences in the resulting photometry naturally arise when using different reduction codes, at least when dealing with HST data.

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However, we highlight that, for this project, the MinnIAC/IAC-pop method presented in Section 4.1 is not sensitive to zero-point systematics, and therefore the impact of small differences in the photometry on the derived SFH is minimized.

The RGB bump is a feature commonly observed in old stellar systems (e.g., Riello et al. 2003), whose physical origin is relatively well understood (Thomas 1967; Iben 1968). When a low-mass star ascends the RGB, the H-burning shell moves outward and reaches the chemical discontinuity left by the convective envelope at the time of its greatest depth. This forces a rearrangement of the stellar structure and a temporary decrease in luminosity, before the star resumes evolving toward higher luminosities and lower effective temperatures. From the observational point of view, since the star crosses the same luminosity interval three times, it produces an accumulation of objects at one point in the luminosity function of the RGB. Figure 8 shows the observed Cetus RGB luminosity function for both sets of photometry. The sample stars were selected in a box closely enclosing the RGB, in order to minimize the contribution of AGB and HB stars. The RGB bump clearly shows up, and fitting a Gaussian profile indicates a peak magnitude of m_F814∼ 23.84 ± 0.08, with excellent agreement between the two photometry sets. An in-depth analysis of the RGB bump of five LCID galaxies is presented in Monelli et al. (2010b).

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This method has been developed by members of the Instituto de Astrofísica de Canarias (A.A., S.H., and C.G.), and now includes different independent modules. Each software package has been presented in different papers (IAC-star: Aparicio & Gallart 2004; MinnIAC: Hidalgo et al. 2010; IAC-pop: Aparicio & Hidalgo 2009), which are the reference publications for the interested reader and include appropriate references.²⁰ In the following, we give only a practical overview of the parameters adopted to derive the SFH of Cetus.

The entire procedure can be divided into five steps. Note that in the following we will use "synthetic CMD" to designate the CMD generated with IAC-star, and we will call it a "model CMD" if the observational errors have been simulated in the synthetic CMD.

• 1.  
  The synthetic CMD. It is created using IAC-star (Aparicio & Gallart 2004). We calculated a synthetic CMD with 8,000,000 stars from the inputs described in the following. We have verified that, with this large number of stars, no systematic effects in the derived SFH of mock galaxies—such as those discussed in Nöel et al. (2009)—were observed, and so no corrections such as those described in that paper are necessary. The requested inputs are as follows.

  • (a)  
    A set of theoretical stellar evolution models. We adopted the BaSTI (Pietrinferni et al. 2004) and Padova/Girardi (Girardi et al. 2000) libraries.
  • (b)  
    A set of bolometric corrections to transform the theoretical stellar evolution tracks into the ACS camera photometric system. We applied the same set, taken from Bedin et al. (2005), to both libraries.
  • (c)  
    No a priori age–metallicity relation is adopted: the number of stars is constant as a function of age, in the range 0 < t < 15 Gyr, and of metallicity, for 0.0001 <Z< 0.005. We stress that the age and metallicity intervals are deliberately selected to be wider than the ranges expected in the solution. This is done to ensure that no information is lost, and to provide the code with enough degrees of freedom to find the best possible solution.
  • (d)  
    The initial mass function (IMF), taken from Kroupa (2002). This is expressed with the formula N(m)dm = m^−αdm, where α = 1.3 for stars with mass smaller than 0.5 M_☉, and α = 2.3 for stars of higher mass. Different values of both exponents have been tested with IC 1613 and LGS 3. The results, presented in E. D. Skillman et al. (2010, in preparation), show that the best solutions are obtained with values compatible with the Kroupa IMF.
  • (e)  
    The binary fraction, β, and the relative mass distribution of binary stars, q.²¹ We tested six values of the binary fraction, from 0 to 1, in steps of 0.2, with fixed q> 0.5; in the following analysis, we use 0.4 and discuss the impact of different assumptions in the Appendix.
  • (f)  
    The assumed mass loss during the RGB phase follows the empirical relation by Reimers (1975), with efficiency η = 0.35.
• 2.  
  The error simulation. The code to simulate the observational errors in the synthetic CMD, called obsersin, has been developed following Gallart et al. (1996), and is described in Hidalgo et al. (2010). It takes into account the incompleteness and photometric errors due to crowding using an empirical approach, taking the information from the completeness test described in Sections 2.2 and 2.4. This is a fundamental step because the distribution of stars in the observed CMD is strongly modified from the actual distribution due to the observational errors, particularly at the faint magnitude level near the old TO, which is where most of the information on the most ancient star formation is encoded. Figure 9 shows an example of synthetic CMD as generated by IAC-star (upper panel), and after the dispersion according to the observational errors of Cetus (lower panel).
• 3.  
  Parameterization. MinnIAC is a suite of routines developed specifically to accomplish two main purposes: first, an efficient sampling of the parameter space to obtain a set of solutions with different parameterizations of the CMD, and second, estimation of the average SFH from the different solutions that best represent the observations. In our case, "parameterization" of the CMD means two things: (1) defining the age and metallicity bins that fix the simple populations and (2) defining a grid of boxes covering critical evolutionary phases in the CMDs, where the stars of the observed and model CMD are counted.The basic age and metallicity bins defining the simple populations used in this work are as follows.Age: [1.5 2 3... 14 15]*10⁹ years.Metallicity: [0.1 0.3 0.5 0.7 1.0 1.5 2.0]*10⁻³.Note that model stars younger than 1.5 Gyr were not included because there is no evidence of their presence in Cetus. These numbers define the boundaries of the bins, not their centers. Therefore, they fix 14 × 6 = 84 simple populations. The choice of a 1 Gyr bin size at all ages was adopted after testing a range of bin widths. It was found that smaller bins do not increase the age resolution; rather they increase the noise in the solution. The selected size of 1 Gyr is the optimal compromise to fully exploit the data, within the limits imposed by the observational errors and the intrinsic age resolution in the CMD at old ages.The parameterization of the CMD relies on the concept of bundles (see Aparicio & Hidalgo 2009 and Figure 10), that is, macro-regions on the CMD that can be sub-divided into boxes using different appropriate samplings. The number of boxes in the bundle determines the weight that the region has for the derived SFH. This is an efficient and flexible approach for two reasons. First, it allows a finer sampling in those regions of the CMD where the models are less affected by the uncertainties in the input physics (MS and SGB). Second, the dimensions of the boxes can be increased, and the impact on the solution decreased, in those regions where the small number of observed stars could introduce noise due to small number statistics or where we are less confident in the stellar evolution predictions. We optimized the values for this work using four bundles, with the following box sizes. The smallest boxes (0.01 mag × 0.2 mag, N1 ∼ 900 boxes) sample the lower (old) MS, and the SGB (bundle 1). The region of the candidate BSs (bundle 2) is mapped with slightly bigger boxes (0.03 mag × 0.5 mag, N2 ∼ 90 boxes), to minimize the number of boxes with zero or few stars. Bundle 3 samples the region between the BS sequence and the RGB. No bundles include the stars in evolved evolutionary phases, and in particular we excluded both the RGB and the HB from our analysis. The reasons are many. First, the physics governing these evolutionary phases is more uncertain than that describing the MS, and differences between stellar libraries are more important in these evolved phases (Gallart et al. 2005). Moreover, the details of the HB morphology also depend on highly unknown factors like the mass loss during the RGB phase. However, despite not including the RGB stars, we routinely adopt a bundle that is redder than the RGB (bundle 4 in Figure 10). In order to aid in setting a mild constraint on the upper limit of the metallicity of the RGB Cetus stars, this was drawn at the red edge of the RGB. It includes metal-rich stars present in the model, but very few objects of the observed CMD. Both bundles 3 and 4 contain boxes of 1.5 mag × 1.0 mag (N3 = 7, N4 = 4). Extensive tests have shown that including a bundle on the RGB neither improves the solution, nor produces a solution significantly different. Rather, the χ² of the solution increases substantially and likely spurious populations appear, such as old and metal-rich stars.For a given set of input parameters, the simple populations plus the grid of bundles "+" boxes, MinnIAC counts the stars in the boxes for both the observed and all the simple populations in the model CMD. A simple table with these star counts is the input information to run IAC-pop. However, our numerous tests disclosed that this approach can be significantly improved by adopting a series of slightly different sets of input parameters, used to derive many solutions and then calculating the mean SFH. This "dithering approach," which is the distinctive feature of the code, significantly reduces the fluctuations associated with the adopted sampling (both in terms of simple populations and boxes) of each individual solution (Hidalgo et al. 2010).In the case of the Cetus SFH, the age and metallicity bins are shifted three times, each time by an amount equal to 30% of the bin size, with four different configurations: (1) moving the age bin toward increasing age (with fixed metallicity), (2) moving the metallicity bin toward increasing metallicity (with fixed age), (3) moving both bins toward increasing values, and (4) moving toward decreasing age and increasing metallicity. These 12 different sets of simple populations are used twice, shifting the boxes a fraction of their size across the CMD.Moreover, to take into account the uncertainties associated with the distance and reddening estimates, together with all the hidden systematics possibly affecting the zero points of the photometry, MinnIAC also repeats the whole procedure after shifting the observed CMD (not the model) a number of steps in both color and magnitude. The bundles are correspondingly shifted. For this project, we adopted an initial grid of 25 positions, shifting the observed CMD in magnitude by [−0.15, −0.075, 0, +0.075, +0.15] mag, and in color by [−0.06, −0.03, 0, +0.03, +0.06] mag. Since we calculated 24 solutions for each node of this grid, we derived a total of 600 solutions.We stress that the position of the best χ²_ν in the magnitude and color grid is not intended for estimates of distance or reddening, since photometry zero points or model systematics also play a role. However, we think it is reasonable to take the best agreement between model and observations in that particular point of the (δ_col, δ_mag) grid as the pointer of the best "absolute" SFH solution.
• 4.  
  Solving. IAC-pop (Aparicio & Hidalgo 2009) is a code designed to solve the SFH of a resolved stellar system. It uses only the information of the star counts in the different boxes across the CMD. Therefore, it calculates which combination of partial models gives the best representation of the stars in the observed CMD, using the modified χ²_ν merit function introduced by Mighell (1999). Note that IAC-pop solves the SFH using age and metallicity as independent variables, without any assumption on the age–metallicity relation. IAC-pop was run independently on each different initial configuration created with MinnIAC.
• 5.  
  Averaging. A posteriori, MinnIAC was also used to read the individual solutions, and calculate the mean best SFH.

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The basis of the MATCH method of measuring SFHs (Dolphin 2002) is that it aims at minimizing the difference between the observed CMD and synthetically generated CMDs. The SFH that produces the best-fit synthetic CMD is the most likely SFH of the observed CMD. To determine the best-fit CMD, the MATCH method uses a Poisson maximum-likelihood statistic to compare the observed and synthetic CMDs. For specific details on how this is implemented in MATCH, see Dolphin (2000a).

To appropriately compare the synthetic and observed CMDs, we binned them into Hess diagrams by 0.10 mag in m_F475W and 0.05 mag in m_F475W − m_F814W. For consistency, we chose parameters to create synthetic CMDs that closely matched those used in the IAC-pop code presented in this paper:

• 1.  
  a power law IMF with α = 2.3 from 0.1 to 100 M_☉;
• 2.  
  a binary fraction β = 0.40;
• 3.  
  a distance modulus of 24.49, A_F475W = 0.11; and
• 4.  
  the stellar evolution libraries of Marigo et al. (2008), which are the basic Girardi models with updated AGB evolutionary sequences.

The synthetic CMDs were populated with stars over a time range of log t = 6.6 to log t = 10.20, with a uniform bin size of 0.1. Further, the program was allowed to solve for the best-fit metallicity per time bin, drawing from a range of [M/H] = −2.3 to 0.1, where M canonically represents all metals in general. The depth of the photometry used for the SFH was equal to 50% completeness in both m_F475W and m_F814W.

To quantify the accuracy of the resultant SFH, we tested for both statistical errors and modeling uncertainties. To simulate possible zero-point discrepancies between isochrones and data, we constructed SFHs adopting small offsets in distance and extinction from the best-fit values. The rms scatter between those solutions is a reasonable proxy for uncertainty in the stellar evolution models and/or photometric zero points. Statistical errors were accounted for by solving 50 random realizations of the best-fit SFH. Final error bars are calculated summing in quadrature the statistical and systematic errors.

The third method we applied to measure the SFH of Cetus is a simulated annealing algorithm applied to our DOLPHOT photometry and the Padova/Girardi library. The code is the same as that applied to the galaxies Leo A (Cole et al. 2007) and IC 1613 (Skillman et al. 2003), and details of the method can be found in those references. The technique treats the task as a classic inverse problem by finding the SFH that maximizes the likelihood, based on Poisson statistics, that the observed distribution of stars in the binned CMD (Hess diagram) could have been drawn from the model.

The code uses computed stellar evolution and atmosphere models that give the stellar colors and magnitudes as a function of time for stars of a wide range of initial masses and metallicities. The isochrone tables (Marigo et al. 2008) for each (age and metallicity) pair are shifted by the appropriate distance and reddening values, and transformed into a discrete color–magnitude–density distribution ϱ by convolution with an IMF (see Chabrier 2003, their Table 1) and the results of the artificial-star tests simulating observational uncertainties and incompleteness. The effects of binary stars on the observations are simulated by adopting the binary star frequency and mass ratio distribution from the solar neighborhood, namely, 35% of stars are single, 46% are parameterized as "wide" binaries, i.e., the secondary mass is uncorrelated with the primary mass, and is drawn from the same IMF as the primary, and finally 19% of stars are parameterized as "close" binaries, i.e., the probability distribution function of the mass ratio is flat. The progeny of dynamical interactions between binaries (e.g., blue stragglers) is not modeled. The history of the galaxy is then divided into discrete age bins with approximately constant logarithmic spacing in order to take advantage of the increasing time resolution allowed by the data at the bright (young) end. The SFH is then modeled as the linear combination of ψ(t, Z) that best matches the data by summing the values of ϱ at each point in the Hess diagram. The best fit is found via a simulated annealing technique that converges slowly on the maximum-likelihood solution without becoming trapped in local minima (such as those produced by age–metallicity degeneracy).

In our solution, we considered only the DOLPHOT photometry. The critical distance and reddening parameters were held fixed at (m − M)₀ = 24.49 and E(B − V) = 0.03. We used 10 time bins ranging from 6.60 ⩽ log(age/Gyr) ⩽10.18. Isochrones evenly spaced by 0.2 dex from −2.3 ⩽ [M/H] ⩽ −0.9 were used, considering only scaled-solar abundances but with no further constraints on the age–metallicity relation. The CMD was discretized into bins of dimension 0.10 mag × 0.20 mag, and it is the density distribution in this Hess diagram that was fit to the models. The error bars are determined by perturbing each component of the best-fit SFH in turn and resolving for a new best fit with the perturbed component held fixed. The size of the perturbation is increased until the new best fit is no longer within 1σ of the global best fit.

Figure 11 summarizes the comparison of the solutions calculated using different sets of photometry, stellar evolution libraries, and SFH codes.

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The upper panel shows the cumulative mass fraction, as a function of time, based on the BaSTI library + IAC method applied to the two sets of photometry. The solution based on the DOLPHOT photometry is marginally younger than the DAOPHOT one. It shows slightly lower star formation at older epochs, and a steeper increase around 12 Gyr ago. In either case, Cetus completely stopped forming stars ∼8 Gyr ago.

The middle panel compares the same two curves with analogous ones obtained using the Girardi library. The effect of changing the library introduces a small systematic effect; both solutions with the Girardi library shift more star formation to slightly (⩽1 Gyr) younger ages compared to the solutions using the BaSTI library. This effect seems more evident at the oldest epochs in the case of the DAOPHOT solutions.

The bottom panel shows the results of the three different SFH reconstruction methods, applied to the same photometric catalog and with the same stellar evolution library. Since for the MATCH and COLE method we present the result of the best individual solution, and not the average of many as in the case of the IAC method, the large squares and triangles highlight the edge of the age bins adopted. Note the good agreement between the MATCH and COLE with the IAC solution, in spite of the higher time resolution of the latter.

For simplicity, we will now focus our analysis on the SFHs obtained with the IAC method and the BaSTI stellar evolution library. At this time, we favor using the BaSTI stellar evolution library because the input physics has been more recently updated, in comparison to the Girardi stellar evolution library (Pietrinferni et al. 2004). Similarly, the IAC method will be used preferentially because it was tailored to the needs of this project and affords a greater degree of flexibility for analysis (Hidalgo et al. 2010).

5.1.1. Exploring the Zero-point Systematics

Figure 12 shows the grid of color and magnitude shifts adopted to derive the IAC-pop solution. The plus signs mark the positions of the 25 initial points. In each of these 25 positions, we calculated the mean χ²_ν, averaging the χ²_ν of the 24 individual solutions. This allows us to study how the χ²_ν varies as a function of the magnitude and color shifts. In particular, the minimum averaged χ²_ν,min identifies the position corresponding to the best solution, which is calculated by MinnIAC averaging the corresponding 24 individual solutions.

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Because the initial color and magnitude shifts are relatively large, we calculated more solutions on a finer grid with 14 positions around the identified minimum χ²_ν (Figure 12, full circles and open squares for the DAOPHOT and DOLPHOT solution, respectively). Therefore, we obtained a total of (25 + 14)*24 = 936 solutions. This computationally expensive task was made possible by the Condor workload management system (Thain et al. 2005)²² available at the Instituto de Astrofísica de Canarias. In the case of the DAOPHOT photometry, the minimum was confirmed at the position (δ_col; δ_mag) = (0.0; 0.075), χ²_ν,min = 1.95, while in the case of the DOLPHOT photometry the minimum is located at (δ_col; δ_mag) = (+0.03, 0.075), χ²_ν,min = 2.88. The two curves around the minimum position of Figure 12 represent the 1σ confidence area around the minimum, calculated as the dispersion around the mean of the 24 χ² of the individual solutions (Aparicio & Hidalgo 2009). Note that the color difference between the χ²_ν,min of the two photometry sets is fully consistent with, and apparently compensates for, the color shift between the two photometry sets (∼0.04 mag). This indicates that the flexibility of our method is able to deal with subtle photometric calibration systematics, and possible stellar evolution model, distance, and reddening uncertainties.

Figure 13 shows a direct comparison between the 24 individual DAOPHOT and DOLPHOT solutions at the χ²_ν,min position. The horizontal lines show the sizes of the age bins, and do not represent error bars. It is noteworthy that the spread of the 24 solutions, derived using different input samplings, is quite small, suggesting the robustness of the mean SFH. Note also the good agreement between the solutions from the two photometry sets. The general trend is the same, with the main peak of star formation occurring within ∼0.5 Gyr and, most importantly, characterized by the same duration.

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5.1.2. The Age of the Oldest Stars

Another important aspect to be addressed is the occurrence of star formation at epochs older than 14 Gyr. As discussed in Section 4.1, we did not impose any strong constraints on the ages of the oldest stars in the model used to derive the SFH of Cetus. That is, we let the code search for the best solution using model stars as old as 15 Gyr, significantly older than the age of the universe commonly accepted after the Wilkinson Microwave Anisotropy Probe (WMAP) experiment (Bennett et al. 2003). Interestingly enough, our solution shows very low SFR for epochs older than 14 Gyr, with a steady increase toward a peak at ∼12 Gyr.²³

To further investigate the significance of deriving ages of stars older than ≃14 Gyr, we performed the following test. Using the derived SFH, we calculated a synthetic CMD, from which we removed all the stars older than 13.5 Gyr. We simulated the observational errors and then derived the SFH of this new CMD identically as for Cetus. Figure 14 presents the input and recovered ψ(t). The latter shows some level of star formation at epochs older than 13.5 Gyr, when there should be none. This means that the combination of observational errors and the uncertainties associated with the SFH algorithms is responsible for the smearing of the main peak of star formation at the oldest epochs.

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Therefore, we decided to adopt the external constraint on the age of the universe coming from the cosmic microwave background experiments, and we set the upper limit to the model stars' age to 13.5 Gyr. This choice is also supported by the fact that current stellar evolution models give Milky Way globular cluster ages in very good agreement with this estimated age of the universe (Marín-Franch et al. 2009). The effect of this constraint is shown in Figure 15. The main effect of limiting the age of the oldest stars in the synthetic reference CMD is that the peak of star formation is higher (since total star formation must be conserved). However, there is no effect on the age or the end of the main episode of star formation, nor on the star formation at epochs younger than 10 Gyr. With both models, we calculate that Cetus completely stopped forming stars ∼8 Gyr ago. This test was repeated with both the DAOPHOT and DOLPHOT solutions, leading to the same conclusion. Thus, we adopt this constraint on the SFH for the rest of our analysis.

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The solutions for each set of photometry, calculated as the average of the 24 solutions at the χ²_ν,min, are represented in Figure 16 where a three-dimensional smoothed histogram summarizes the main features. This is a useful representation giving an overall view of the SFH of a system (Hodge 1989), including the SFR as a function of time, the metallicity distribution function, and the age–metallicity relation. It shows graphically how the oldest stars are more metal poor, and how chemical enrichment proceeds toward younger ages. The two projections ψ(z) and ψ(t) are the metallicity distribution integrated over time (blue line) and the age distribution integrated over metallicity (red line). Errors have been calculated for ψ(t) and ψ(z) as the dispersion of the 24 individual solutions used in the average, plus the contribution of the statistical errors (see Aparicio & Hidalgo 2009). Figure 16 shows, that, in particular, the ψ(z), presents a peak at the most metal-poor regime, and a sharp decrease toward the more metal-rich tail. Only a negligible fraction of stars ever formed have metallicities higher than Z = 10⁻³.

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The overall agreement between the DAOPHOT and the DOLPHOT solutions, illustrated in Figures 13 and 16, suggests that we can safely adopt the average of the two solutions for the SFH of the Cetus dSph galaxy, and the difference of the two as an indication of external errors.²⁴

Figure 17 summarizes the main results by comparing the SFR as a function of time (top), the age–metallicity relation (middle), and the cumulative mass function (bottom), for both sets of photometry with the adopted average calculated as explained in Hidalgo et al. (2010). The thick dashed and continuous lines represent the DAOPHOT and DOLPHOT solutions, while the thin lines are the corresponding error bars, which include both internal and statistical errors, calculated following the prescriptions given in Hidalgo et al. (2010). The thick gray line is the average of the solutions obtained for the two photometries. Figures 16 and 17 provide two complementary ways of presenting the derived SFH of Cetus. From these figures, we see that Cetus is mostly an old, metal-poor stellar system, with the vast majority of stars older than 8 Gyr and more metal poor than Z = 0.001. Table 1 summarizes the main integrated quantities derived for both photometry sets.

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The vertical lines in the top panel indicate the redshift z = 15 and z = 6, which mark the epoch when reionization was under way, the latter considered to be the epoch when the universe was fully reionized (Bouwens et al. 2007 and references therein). The vertical lines in the middle panel of Figure 17 mark the epoch when 10%, 50%, and 90% of the stellar mass was formed. They indicate that Cetus experienced the bulk of its star formation before redshift ≈2, with a peak at z ∼ 4.

In the following, we focus on two important aspects of the Cetus SFH: the sequence of candidate BSs and the actual duration of the main episode of star formation.

5.2.1. Recent Star Formation or Blue Stragglers

In Section 3, we gave circumstantial evidence, based on the comparison with isochrones, that the objects bluer and brighter than the old MSTO are BSs, rather than truly young stars. More evidence of this comes from the analysis of the SFH presented in Figure 16. The plot shows an event of very low SFR that produced a small percentage of metal poor but relatively young (∼2–4 Gyr old) stars. This population clearly does not follow the general age–metallicity relation. Figure 18 shows that the position of these stars in the calculated CMD is fully consistent with the position of the blue plume on the CMD. This feature is systematically present in all our solutions, and does not depend on the sampling, library, or photometry adopted. Given the age–metallicity relation derived for Cetus, there is no reason to expect a relatively young but very metal poor population—this would require something like an episode of late infall of very metal poor gas. This interpretation is nicely supported by the age–metallicity relation presented in Figure 17 (middle panel), which shows a continuous increase of metallicity with time, until a maximum occurs 8 Gyr ago. After that, we observe a sharp decline followed by a renewed increase. If we accept that BSs are found in all or nearly all sufficiently populous open and globular star clusters as well as in the Milky Way field halo population, then there is no conflict between our data and the assertion that no star formation occurred in Cetus in the last 8 Gyr; the most natural explanation for these stars is that they are BSs belonging to the old, metal-poor population. Note that the contribution of this population to the total estimated mass of Cetus is ⩽3%. Therefore, the presence of fainter BSs contaminating the TO and MS regions is expected to have negligible impact on the derived SFH. An in-depth discussion of the candidate BSs in both dSph galaxies of the LCID sample will be presented in a forthcoming paper (M. Monelli et al. 2010, in preparation).

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5.2.2. The Short First Episode of SFH in Cetus

The description presented in Section 5 corresponds to the SFH of Cetus as derived using the discussed algorithms. It is clear that this solution is the result of the convolution of the actual Cetus SFH with some smearing produced by the observational errors, and the limitations of the SFH recovery method (Aparicio & Hidalgo 2009). In the following, we will try to further constrain the features of the actual SFH of Cetus through some tests with mock stellar populations.

To investigate the actual duration of the dominant episode of star formation, and to assess the age resolution at the oldest ages, we performed the following test. We created mock galaxies characterized by fixed metallicity and short episodes of star formation around the mean value of 12.0 Gyr (see Figure 19). Since a Gaussian profile fit to the Cetus ψ(t) yields an estimate of σ = 1.53 Gyr, we used three ψ(t) modeled by a Gaussian profile with σ = 0.5, 1.0, and 1.5 Gyr. We simulated the observational errors in the mock stellar populations, and recovered their SFH using the same prescriptions as for the real data. For simplicity, only the completeness tests of the DAOPHOT photometry were adopted to simulate the observational errors in the mock CMD. The number of stars in the mock populations was such that the total number of stars in the bundles was comparable to the case of the real galaxy.

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The first important result of this test is that the age of the peak of the recovered mock population is systematically well recovered, within 0.2 Gyr. Thus, if the initial episode of star formation in Cetus is well represented by a Gaussian profile, then we are confident in our ability to determine the time of the peak SFR. Similar tests have been also discussed in Aparicio & Hidalgo (2009) and Hidalgo et al. (2010). The second finding is that the duration of the recovered ψ(t) is wider than the input one. In particular, we measured the half-width at half-maximum (HWHM) and calculated the corresponding σ, which are equal to 1.44, 1.61, and 1.95 Gyr for the three episodes of increasing duration. The increase from the input σ is a consequence of the blurring effect of the observational errors and the limited age resolution at old ages.

The comparison of the duration of the main peak in the Cetus and in the mock stellar populations suggests that the estimated duration for Cetus is an upper limit. Moreover, we note that its value can be further constrained by the σ = 0.5 and the σ = 1.0 Gyr Gaussian profile tests, with a result close to σ = 0.8 Gyr (interpolated value). Moreover, we can expect that the limited age resolution broadens both the older and the younger sides of the ψ(t) around its maximum. This is supported by Figure 19, where we can compare the youngest epoch of star formation in the three models and in their recovered profiles. The comparison reveals that the output ψ(t) is more extended to younger ages than the input ones, and that this effect is less important for increasing σ_in. This suggests that at least part of the star formation between 10 and 8 Gyr ago derived in Cetus is not real. However, we stress that the differences between the actual SFH and the recovered one affect only the duration of the star formation solution and the height of the main peak, but they do not alter significantly the age of the peak itself.

Finally, Figure 20 presents a comparison between the observed DAOPHOT CMD and a synthetic CMD corresponding to the best solution (left and central panels). We find a satisfactory agreement in all the main evolutionary phases. As far as the differences in the HB morphology are concerned, it is worth noting that we have not tried to properly reproduce this observational feature. Due to the strong dependence of the HB morphology on the mass-loss efficiency along the RGB, it is clear that a better agreement between the observational data and the synthetic CMD could be obtained with a better knowledge of mass-loss efficiency and its spread. The right panel shows the residual Hess diagram, which supports the good agreement between the observed and the calculated CMD.

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One of the main drivers of the LCID project is to find out whether cosmic reionization left a measurable imprint on the SFHs of nearby, isolated dwarf galaxies. The universe is thought to have been completely reionized before z ⩾ 6 (Bouwens et al. 2007), corresponding to T ≃ 12.8 Gyr ago. The comparison of these values with the Cetus ψ(t) in Figure 17 (see upper panel) shows that the vast majority of the measured star formation in Cetus occurred at epochs more recent than 12.8 Gyr ago (see also Table 2). This is also true when the actual Cetus SFH is considered, as discussed in Section 5.2.2. Therefore, we conclude that there is no evident coincidence between the epoch when reionization was complete and the time in which the star formation in Cetus ended.

However, one might ask the question of whether acceptable solutions could be found that do not show star formation more recent than 12.8 Gyr ago, i.e., that are compatible with the hypothesis that reionization contributed to the suppression of star formation in Cetus. With this question in mind, we performed two additional tests, summarized in Figure 21.

• 1.  
  We calculated a solution of the SFH of Cetus constraining the model CMD to stars of ages older than 12.8 Gyr. This is not the typical approach for the IAC method, but nonetheless the code looks for the absolute minimum in the allowed parameter space, and provides the best possible solution (gray dashed line). However, we found that this solution is clearly at odds with the derived best SFH of Cetus, and not compatible with it. Also, note the significantly higher χ²_ν, which increases to 7.80, and the large residuals (Figure 21, central panel). The latter, calculated with respect to the observed CMD identically as in Figure 20, indicate important systematics in the solution. The excess of stars in the calculated CMD in the red part of the TO (lightest bins), indicates the presence of too many stars in this region. This is in agreement with the overall older population, with the main peak at ∼13 Gyr ago.
• 2.  
  We recovered the SFH of a mock population, similar to the tests shown in Figure 19, created with constant SFR in the age range between 13 and 13.5 Gyr, and zero elsewhere. The solution shown in Figure 21 discloses that the peak is recovered at the correct age, in the oldest possible age bin (gray solid line). This suggests that an actual SFH where most of stars were born before the end of the reionization epoch is not compatible with the solution we derived for Cetus. Moreover, the general trend in the residual plot (Figure 21, right panel), is similar to the previous case, with an excess of red TO stars.

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These two tests, together with the ones shown in Figure 19, suggest that, on the basis of the available data coupled with the SFH recovery code, we can reject the hypothesis that the majority of the star formation occurred in Cetus in epochs older than z ≃6.

The various pieces of information analyzed in Section 5 can be summarized as follows. The star formation of the observed area of Cetus shows a well-defined peak that occurred ≈12 Gyr ago. By resolving the star formation of mock stellar populations, we showed that the actual duration of the main event is probably overestimated due to the smearing effect of the observational errors. This does not affect the age of the main peak, but only its duration, at both older and younger epochs. Therefore, the last epoch of star formation, estimated at 8 Gyr ago, is probably actually somewhat older. The various evidence indicating that the blue plume of objects identified in the CMD consists of a population of BSs, supports the idea that no star formation occurred in Cetus in the last 8–9 Gyr. In Table 2 we present the epoch when various fractions of the total mass were formed, once we excluded the contribution of the BS population. However, the actual values of the percentiles reported could be slightly different due to the smearing of the solution with respect to the actual, underlying SFH (see Section 5.2.2).

The derived SFH is at odds with the conclusion by Sarajedini et al. (2002), who suggested that Cetus might be 2–3 Gyr younger that the old Galactic globular clusters. The stability of the main peak around the age of 12 Gyr, together with up-to-date estimates of the globular cluster ages (Marín-Franch et al. 2009), suggests that such a difference can be ruled out.