This Appendix is meant to give a more elaborate description of the data reduction steps (Sect. A.1) and in particular the procedure for homogenising the PSF and measuring colours using Gaussian weighted apertures (Sect. A.2). Because we combine photometric data over a wide range of wavelengths and for clusters that are both in the northern and southern sky, we necessarily have to combine data from different telescopes and/or instruments.
The standard reduction steps include bias and flatfield corrections. Although the images have been flatfielded (e.g. by Elixir for the MegaCam data) to yield a uniform zeropoint for the source fluxes, there are still residual background patterns due to scattered light, fringe residuals, and amplifier drift (Cuillandre, priv. comm.). These patterns are reasonably stable over time, and since most exposures in a given filter have been taken consecutively on the same night, we can subtract these background effects. We do this by using the dithered pattern of observations to differentiate signals that are on a fixed position on the ccd array from sky-bound signals.
To remove cosmic rays from ccd images one usually compares different frames of the same part of the sky. However, since we only have a few deep exposures in some of the filters, the number of overlapping frames of our data set is not always sufficient to be able to identify all cosmic rays. Therefore we remove cosmic rays by using the Laplacian Cosmic Ray Identification method (van Dokkum 2001), which works on individual images. We optimise the parameters in the setup of the code such that we minimise the amount of false positives (bright stars) and false negatives. We do this by testing the code on a range of images with different seeing. The only parameter that has a significant influence on the fraction of false positives and false negatives is objlim, which we take to be 3.0.
We obtain astrometric and relative photometric solutions for each chip using SCAMP (Bertin 2006), where we use all exposures in a given filter for all clusters at once to effectively increase the source density, and obtain stable solutions. As a reference catalogue we use SDSS-DR7 data, or the USNO-B catalogue whenever a cluster falls outside the SDSS footprint. This leads to consistent astrometric solutions between the different filters with an internal scatter of about 0.05′′.
GCLASS photometric data set.
Because the shape and size of the PSF are different between exposures and filters, it is non-trivial to measure accurate colours of a galaxy. The simplest approach would be to take the ratio of the total flux of a galaxy in different bands, but this requires very large photometric apertures: for background-limited observations these are very noisy.
However, a reliable colour measurement for the purpose of photometric redshift determination can also be made by taking the ratio of aperture fluxes in different bands, provided these apertures represent the same intrinsic part of the galaxy. We have followed this approach here, based on a modification of the Gaussian-aperture-and-PSF (GaaP) photometry method (Kuijken 2008).
The first step is to convolve each image with a suitable position-dependent kernel that modifies its PSF into a uniform size, circular and Gaussian. This kernel can be constructed using the shapelet (Refregier 2003; Kuijken 2006) formalism, as was done in the local approach described in Hildebrandt et al. (2012), with one modification: here we allow the resulting PSF size for each image to be different. Specifically, for each filter and field we choose the size of the resulting PSF to be slightly larger (ca. 10%) than the median gaussian radius of all bright stars in the images. To obtain a stable PSF in the stacked images for each filter we Gaussianise the PSF of the individual astrometrically corrected exposures before stacking.
Following Kuijken (2008) we then measure fluxes in the following way. Instead of using a function where the weight is either 0 or 1, as is the case for regular aperture photometry measured with a top-hat weighting function (e.g. by running SExtractor in dual image mode), we use a smooth weight function that makes use of the fact that the S/N for each pixel decreases away from the peak pixel. When the PSF in each filter follows a Gaussian profile, the choice to perform photometry using a Gaussian weight function is computationally convenient, as we show next.
Kuijken (2008) defines the “Gaussian-aperture-and-PSF” flux Fq as the Gaussian weighted flux a source would have if it were observed with a Gaussian PSF with the same width q as the weight function. Hence (A.1)where S is the intrinsic light distribution of the source (i.e. before smearing with the PSF) and q is the scale radius of the weight function. It is straightforward to simplify Eq. (A.1) to (A.2)which shows that Fq is a Gaussian-aperture photometric measurement of the intrinsic galaxy.
After gaussianising the images, S has already been convolved with a Gaussian that has a constant dispersion gPSF for each stacked image. The flux distribution on the ccd is therefore (A.3)Analytically we have an identical expression for Fq(A.4)which thus shows that the same intrinsic aperture flux Fq can be measured from images with a range of Gaussian PSF sizes. Therefore, from our PSF-gaussianised images, we can measure colours of the same intrinsic part of the galaxy if we use Gaussian weight functions to measure fluxes. Note that it is no longer necessary that the stacks of the different filters have a PSF with the same Gaussian FWHM, as long as the weight function is adapted accordingly for each filter.
We adjust q to ensure the aperture roughly matches each galaxy’s size, to optimise the S/N. We base our choice for q on the SExtractor parameter FLUX_RADIUS measured in the Ks-band image, such that q = 0.85·FLUX_RADIUS. The factor of 0.85 is chosen to optimise the S/N of a source with a circular Gaussian PSF-profile. Further we make sure that q is chosen such that q > gPSF in all filters.
This method is applied to measure fluxes in the u − Ks-bands, but since the IRAC data suffer from a much larger PSF we work in a two stage process to incorporate the IRAC fluxes in a way that reduces the problems from confusion. We construct a 2-stage multi-colour catalogue where we multiply the IRAC flux measured in the bigger aperture with the fractional difference of the Ks-band flux measured in the small and bigger aperture. This way we effectively correct the IRAC flux for blending with nearby objects by assuming these neighbours have the same (Ks-IRAC) colour as the source. For contaminating galaxies this is often the case. To verify that any residual blending in the IRAC bands does not affect our results, we repeated the analysis while excluding the IRAC data in all SED fits. We find no bias in the stellar mass estimates, and even for the lowest masses (M⋆ = 1010 M⊙), 68% of the estimated stellar masses differ by less than 0.05 dex from our fiducial analysis.
We calibrate the photometric zeropoints on a catalogue basis by making use of the universality of the stellar locus (High et al. 2009). We use stellar data from Covey et al. (2007), containing 600 000 point sources selected from the SDSS and 2MASS surveys. By applying linear colour terms we compare these colours to stars measured with the filter sets in the telescope we used. Note that these data are especially favourable to calibrate the zeropoints using the stellar locus since the amount of galactic dust is very low in these fields. We adapt the zeropoints of the ugrizJKs filters to bring the colours of stars in our data in line with the reference catalogue. Corrections are typically of the order of 0.05 magnitudes. To account for uncertainties in the absolute zeropoint of IRAC, we included a 10% systematic error to the IRAC fluxes.
After gaussianisation, the background noise in the images is correlated between pixels. Therefore we estimate the errors on the flux measurements in the stacks of each filter by measuring the fluctuations in the flux values measured from apertures that are randomly placed on the images. We take account of the non-uniform exposure time over the image stacks. Table A.1 shows an overview of the median 5-σ flux measurements for point sources in each filter and each cluster.
UltraVISTA (magenta) versus GCLASS (black) field measurements. Left panel: total galaxy population in both fields. Middle panel: SMF for the sub-set of star-forming galaxies. Right panel: sub-set of quiescent galaxies. Error bars show the 68% confidence regions for Poisson error bars. The grey curves show the 10 contributions to the field SMF around the GCLASS clusters, which differ because of cosmic variance due to the small volumes probed in these individual fields. Also the fields contribute only down to a particular mass respecting the varying depths of the GCLASS fields. Bottom panels: fractional differences between the two field measurements, given by , together with the estimated errors.
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Thanks to the relatively wide areas that were observed to obtain the GCLASS multi-colour catalogues (15′ × 15′ centred on the clusters in the northern sky, and 10′ × 10′ for the clusters in the southern sky), these data can also be used to study galaxies outside the clusters and hence to measure the SMF of the general field. In this appendix we measure the field SMF from GCLASS in the redshift range 0.85 < z < 1.20 and compare this to the field SMF measured from UltraVISTA.
Since the UltraVISTA sample is based on a relatively deep (compared to GCLASS) 30-band photometric catalogue, it is complete in the mass range (M⋆ > 1010 M⊙) at this redshift range. A comparison between GCLASS and UltraVISTA may reveal possible systematic differences in the stellar mass catalogues, and any residual incompleteness in GCLASS.
To minimise the contamination by cluster galaxies in the sample, we use a conservative selection of field galaxies in GCLASS. A galaxy is considered as part of the field when it is separated from the cluster centre by more than the angular distance that corresponds to 1.5 Mpc at the redshift of the cluster. Furthermore we require a field galaxy to have a photometric redshift | zphot − zcluster | > 0.05. After taking account of the areas masked by bright stars, this results in a total probed volume of the field that is ~6 times smaller in GCLASS compared to UltraVISTA. Since the 10 GCLASS pointings have different depths, we have to take account of the estimated mass-completeness of the detection bands. This is measured similarly as Sect. 3.2, but using a redshift limit of 1.20 in each field (instead of the individual cluster redshifts). This way we correct for Malmquist bias in a similar way as in the 1/Vmax weighting method.
Figure A.1 shows a comparison of the field SMF measured in the GCLASS (black) and UltraVISTA (magenta) surveys. The curves are normalised with respect to the total volume subtended by these surveys. The grey curves show the contributions to the field SMF of the 10 individual GCLASS fields. These contributions differ between the pointings because their depths are different, and also the area surrounding the cluster that is part of the field differs. The differences in the grey curves are further caused by cosmic variance. The field in the SpARCS-1047 image for example is significantly overdense in the redshift bin 0.85 < z < 1.20. Note, however, that, when these 10 independent sight-lines of GCLASS are combined, the uncertainty due to cosmic variance is greatly reduced (Somerville et al. 2004).
There is generally a good agreement between the field SMF measurements from GCLASS and UltraVISTA, especially at the high-mass end. This indicates that there are no substantial systematic differences between the two catalogues this study is based on. At the low-mass end of the SMF there are some systematic differences in both the star-forming and quiescent population, increasing to several ~10% in the lowest mass bins. In Sect. 4.1 we explained that we corrected the GCLASS cluster
SMF data by these completeness correction factors. That way we cannot only compare the cluster and field qualitatively, but have a more realistic view on the absolute Schechter parameters. Note that this additional completeness correction does not change any of the qualitative statements in this paper, nor affects the conclusions of this paper in any way.
© ESO, 2013