Rest–frame absolute magnitudes were calculated from the deep ground–based multi–band imaging data. For each project, ground-based imaging in various filter passbands was acquired which turned out to be especially beneficial for the field studies in the FDF and WHDF. The apparent brightness of a given field galaxy was derived from the filterXwhich best matched theB-band response curve in rest–frame. According to the redshift of the respective galaxy, this filterXwas either theB-,g-,R- orI-band (see section4.5.3).
For the E+S0 galaxies the adopted equation for the computation of absolute Y magnitudes was
MY = mX −AX −DMΛ(z, H0)
−kY(X, T, z), (4.8) whereMY denotes either the absoluteB magni-tude (field galaxies) or absolute Gunn r magni-tude (cluster galaxies).
mX is the apparent total brightness in the pass-band X as derived with the Source Extractor (see section4.5.1).
AX is the Galactic absorption in the wavelength regime of filterX (see section4.5.2).
DMΛ(z, H0) is the distance modulus for the re-spective redshift of a cluster or field object. As already stated earlier, throughout this thesis the so–called concordance cosmology (e.g., Spergel et al. 2003) is adopted, i.e. a flat universe with a matter density Ωm = 0.3, a cosmolog-ical constant corresponding to an energy den-sity of ΩΛ = 0.7 and a Hubble constant of H0 = 70 km s−1Mpc−1. In most previous stud-ies also a non-zero cosmological constant was as-sumed. Possible deviations caused due to dif-ferent values of H0 were accounted for when a comparison with the literature was performed.
kY(X, T, z) is the k–correction for the transfor-mation from filter X to the rest–frame Y-band which accounts for three effects: (i) the different response curves of filter X in observer’s frame and filterY (eitherB or Gunnr) in rest–frame, (ii) for the redshifted and “stretched” SED in observer’s frame and (iii) the SED typeT. For all cases, a SED typeT =−5/−2, i.e. an early-type E/S0 galaxy was adopted. The individual correction factorskY have been derived via syn-thetic photometry, which is described in detail in section4.5.3.
In a second independent approach, for the sub-sample of galaxies with HST structural informa-tion, the rest–frame absolute magnitudes were computed using the transformation procedure as discussed in section 4.3.4.
Chapter 4: Photometric Analysis 85
Table 4.15: Galactic extinction coefficients for the poor cluster fields.
Name R.A. Dec. AB BH AB SFD E(B−V) AF702 AR AI
[mag] [mag] [mag] [mag] [mag] [mag]
CL 0849+37 08 49 11 +37 31 09 0.090 0.151 0.035 0.085 0.094 0.068 CL 1701+64 17 01 47 +64 20 57 0.060 0.112 0.026 0.063 0.070 0.050 CL 1702+64 17 02 14 +64 19 53 0.060 0.110 0.026 0.063 0.068 0.050
4.5.1 Apparent Magnitudes
Total apparent magnitudes were derived with the Source Extractor package (SExtractor, Bertin & Arnouts 1996). This program offers different algorithms for photometry of stellar ob-jects and extended sources and a choice between circular or elliptical apertures. Apparent mag-nitudes measured within fixed circular apertures are the best approach for the derivation of galaxy colours. Total apparent magnitudes are not reli-able to compute galaxy colour indices as the ap-parent diameter of the galaxies could deviate be-tween different passband filters. For the deriva-tion of absolute magnitudes it is mandatory to prevent flux losses. For this purpose, variable elliptical apertures yield the best results.
SExtractor offers three algorithms for the usage of elliptical apertures, called Mag iso,Mag best and Mag auto. Tests showed that the latter two routines are consistent to within a few 0.01 mag for early-type galaxies in all bandpasses and for different field and cluster environments.
However, in particular for faint detections, the Mag isoalgorithm systematically resulted in un-derestimated brightnesses. The Mag auto auto-matic aperture routine is based on the “first mo-ment” algorithm by Kron (1980) and designed to best reproduce the total magnitudes of extended sources. This algorithm has been used for both field studies whereas for the cluster galaxies the Mag best routine was applied.
4.5.2 Galactic Absorption
The correction for the extinction due to Galac-tic absorption was performed in using a special program written by the author, called SFDextc.
This program calculates automatically the ex-tinction coefficients for a specificE(B−V) input value based on theCOBE dust maps by Schlegel et al. (1998, hereafter SFD). As a consistency check, results were compared to the Galactic ex-tinction values by Burstein & Heiles (1982, here-after BH) and AV = 3.1·E(B −V). As the dust maps provide a more reliable measurement only the SFD extinction values were adopted.
For comparison, for the Low–LX clusters also the AB coefficients by BH are shown (see Ta-ble 4.15). The uncertainties in the extinction values are E(B−V) = 0.010m.
For the cluster A 2390 an E(B −V) = 0.110m was assumed, resulting in extinction coefficients for the Johnson-Cousins filters of AU = 0.600m, AB = 0.476m and AI = 0.214m, respec-tively. For the HST/WFPC2 F814W filter A814= 0.214m was derived.
The Galactic extinction coefficients for the Johnson-Cousins and the HST F702W filter passbands of the Low–LX cluster fields are sum-marised in Table4.15.
For the Galactic absorption at the coordinates of the FDF, the values which are given in Heidt et al. (2003) were adopted. These coefficients were derived with the formulae given in Cardelli et al. (1989) under the assumption of E(B − V) = 0.018m (derived from BH) and AV = 3.1·E(B −V). For the individual
broadband filters used for the FDF imaging, the resulting respective absorption factors are AU = 0.087m, AB = 0.076m, Ag = 0.062m, AR = 0.041m and AI = 0.035m. For the HST/ACS F814W filterA814 = 0.035mwas com-puted. At the central coordinates of the WHDF of α2000=00h19m59.6s, δ2000=+00◦0401800 (B1950.0), an E(B−V) = 0.025m was adopted, yielding to Galactic extinction coefficients of AU = 0.137m,AB = 0.108m,AR = 0.067m, and AI = 0.049m, AH = 0.014m, andAK = 0.009m. The absorption coefficient for the HST/ACS F814W filter is A814= 0.049m.
4.5.3 K-Correction
The k-correction accounts for the fact that sources which are observed at different redshifts are compared with each other at different rest-frame wavelengths. The transformation between an observed apparent magnitude in a photomet-ric broad passband X to a corresponding rest–
frame magnitude in another broad-band photo-metric passband Y involves the term known as the “k-correction”. Such a transformation is de-fined by
kY(X, T, z) =X(T, z)obs−Y(T,0)rest (4.9) If photometric broad-band information is lim-ited to a few filters, the different wavelength ranges covered by a passband in observer’s frame and rest–frame introduce a strong dependence of the achievable k-correction accuracy on SED type. At redshift z = 0.2, e.g., the difference between SED types E/S0 and Sbc corresponds to a change of ∆kB = 0.4m in the transfor-mation from Bobs to Brest (e.g., Frei & Gunn 1994). For this reason, even a slight misclas-sification can introduce a substantial offset in the derived luminosity if observations are lim-ited to one or two filters only. In contrast to this, the field galaxy projects of the FDF and WHDF greatly benefitted from the multi–band imaging data. With the FDF photometry in B, g, R and I, and the WHDF photometry in B,
Figure 4.24: Transmission curves of the FORSU, B,g,RandIfilters which were used for the imaging of the FDF.
R and I, the filter that best matched the rest–
frameB-band could be used to transform an ap-parent magnitude X into absolute magnitudes MB. For both, rich and poor early-type cluster samples, observations were carried out in the R or I filter bands. At z = 0.2, the observed R, I and the R702, I814 HST filter passbands (cf.
section 4.3.4) are very close to rest-frame Gunn r. For the clusters CL 0849 (z= 0.234), CL 1701 (z = 0.246) and CL 1702 (z = 0.223) the trans-formation Robs → rrest amounts to 0.02m only.
Note that these numbers are less than the cor-rection term for Galactic extinction. Therefore, the advantages of using the Gunnr-band as rest-frame passband instead of the bluer Johnson V orB bands are the smallerk-corrections and the lower corrections of galactic absorption.
In order to ensure the highest possible accu-racy in the transformation between observed–
frame and rest–frame, the kY-corrections were not taken from the literature but computed via synthetic photometry as outlined in B¨ohm (2003). For a given SED of type T at redshift z with wavelength–dependent flux F(T, λ), the magnitude in a passbandX was defined as
mX(T, z) = −2.5 log
RF(T, z, λ)SX(λ)dλ R SX(λ)dλ
Chapter 4: Photometric Analysis 87
Figure 4.25: Transmission curves of the JohnsonUJ, BJ,VJ and CousinsRC,IC filters.
+CX +Csys (4.10) (e.g., Fritze-v. Alvensleben 1989). The par-ameter SX(λ) is the response function of filter X. CX is a constant for the calibration of the respective filter within the corresponding filter system (e.g, the Johnson, Cousins or Gunn filter system). Csysis an optional constant to calibrate between different filter systems. The derivation of the calibration factors CX and Csys is dis-cussed below. Note that Eq. 4.10 yields rela-tive magnitudes which are appropriate for the transformation between different filters or differ-ent systems, but is not reliable for the compu-tation of physical apparent brightnesses in the form given above. For the latter purpose, an additional calibration constant is needed, which can be derived via synthetic photometry with an observed SED of a galaxy with known ap-parent brightness. However, since the synthetic photometry only was executed to derive the k-corrections, i.e., for relative photometry, this cal-ibration was not needed.
In Fig. 4.24, the FORS filters U, B, R, I and Gunn g are shown. This set of filters was used for the imaging of the FDF in the optical regime.
In Fig. 4.25, the filter response curves of the standard Johnson-Cousins system (UJ, BJ, VJ, RC, IC) are displayed. These curves were
re-Figure 4.26: Transmission curves of the Thuan-Gunnu, gandrfilters.
trieved as ascii tables from the ESO webpages.
Fig. 4.26 shows the filter response curves of the Thuan-Gunn filter system (u, g, r) which were retrieved as ascii tables from the CAHA instru-ments webpages. All these filter curves were used for the definition of the respective response func-tionsSX.
For the Johnson system, the constants CX were taken from Allen (1973). They are defined in such a way to yield zero colour terms for an A0V star spectrum in all passbands. To cal-ibrate the FORS filter system, the A0V spec-trum from Pickles (1998) was used and the fine–
tuning of the respective factors CX was per-formed with the equations given in Sinachopou-los & van Dessel (1998). Finally, the constant Csys for transformations between the FORS and the Johnson system was calculated from the colour (Vfors − VJ) (ibid.). For the Cousins RC and IC filters, the factors CR and CI of the FORS calibration were adopted as initial values. The calibration factors were verified through a comparison of the (RJ − RC) and (IJ−IC) colours, assessed for various SED types, to the numbers published in Fukugita et al.
(1995), which yielded consistent values within
<0.03 mag.
Since the FORS g filter has a slightly different shape from the original Thuan-Gunngfilter
def-Figure 4.27: SED template used for the compu-tation of thek-corrections via synthetic photometry.
This is an observed spectrum of an E/S0 galaxy with an age of 12 Gyrs (Kinney et al. 1996). For this plot, the template has been normalised to the flux at 5500 ˚A.
inition, the following more complicated calibra-tion procedure has to be performed. The FORS g filter is broader by ∼90 ˚A and has a ∼130 ˚A higher central wavelength than the Thuan-Gunn gfilter. A first calibration was conducted adopt-ing the transformation by Jørgensen (1994)
ggunn−VJ = 0.503 (BJ −VJ)−0.226. (4.11) Based on the A0V spectrum, the colour (gfors− ggunn) then was computed, which yielded a value of 0.10 mag and thus a difference to the Johnson system of (gfors(A0V)− VJ(A0V)) = −0.13m. Again, the colours were compared through a cross–correlation with thegpassband definitions given by Fukugita et al.
As SED templates for the computation of the k-corrections, the UV/optical spectra published by Kinney et al. (1996) were used. These observed template spectra comprise different morphologi-cal galaxy types of elliptimorphologi-cal, bulge, S0, Sa, Sb, and Sc and starburst galaxies and cover a wave-length range of∼1200–9800 ˚A with a resolution between ∼8 to 10 ˚A. For the synthetic photom-etry described here, only elliptical galaxy tem-plates with an age of 12 Gyrs were used solely.
One of these templates is shown in Fig. 4.27.
For the purpose of this graph, the spectrum was normalised to the flux at 5500 ˚A. No separation
Figure 4.28: SED template used for the com-putation of the k-corrections via synthetic photom-etry. This spectrum was generated via chemically consistent evolutionary synthesis models (M¨oller et al. 2001). For this plot, the template has been nor-malised to the flux at 5500 ˚A.
between elliptical and S0 types was done be-cause both offer similar (optical and UV) spec-tral shapes, Balmer discontinuities (4000 ˚A) and absorption features for λ > 5000 ˚A which are indistinguishable from their broadband optical and near-IR colours.
For internal consistency, the synthetic spectral SED templates by M¨oller et al. (2001) were also applied to compute thek-corrections. These templates were generated with evolutionary syn-thesis models and the authors provide two ver-sions of each template, one including dust red-dening and one without dust. For the synthetic photometry performed here, templates without dust were used solely, since no intrinsic absorp-tion has to be accounted for early-type galaxies in this wavelength regime. The basic advantage of synthetic spectra with respect to observed spectra is of course that no noise is introduced in the photometry. Elliptical galaxy templates with an age of 12 Gyrs were used in order to give a good reproduction of the observed colors of present–day early-type galaxies. One E/S0 tem-plate is shown in Fig.4.28, which was normalised to the flux at 5500 ˚A for plotting purposes. Gas emission was not incorporated in the models, but this has a negligible effect on broad band pho-tometry for very late–type SEDs (T ≥8) only.
To derive the fluxF at wavelength λ, the SEDs
Chapter 4: Photometric Analysis 89
were redshifted according to
F(T, z, λ) = F0(T, λ/(1 +z))
(1 +z) (4.12)
(e.g., Contardo, Steinmetz & Fritze-v. Alvens-leben 1998), where F0 is the flux of the un–
redshifted spectrum. Finally, the transforma-tion from the apparent magnitude of a spectrum of type T (E/S0) at redshift z observed with a FORS/Johnson-Cousins filterX to the Johnson B or Gunn r magnitude mY in rest–frame was performed via
kY(X, T, z) =mX(T, z)−mY(T, z= 0), (4.13) where the two terms on the right hand side were derived according to Eq.4.10. k-corrections were computed with the equation given above for all source filters (FORS and Johnson-Cousins filter system), redshifts (cluster redshift or redshift of the field galaxy) and environments covered by the early-type galaxies. Typical deviations in the k-corrections between the templates of Kinney et al. (1996) and M¨oller et al. (2001) are very small, e.g. ∆kr = 0.03m in the Gunnrfilter, for the SEDs of both ellipticals and S0 galaxies.
To transform to Johnson B rest–frame (cf.
Eq. 4.8), the following filters were used depend-ing on the redshift: Bfors for z < 0.25, gfors for 0.25 ≤ z < 0.55 and Rfors for 0.55 ≤ z < 0.7 (only one object). For this reason, the k-correction was much less sensitive to spectral type than using a global transformation of Bobs → Brest, which was particularly important at higher redshifts z >∼ 0.6 in the case of spi-ral galaxies with late–type SEDs (T ≥ 1) in the FDF (B¨ohm 2003). An analogous approach was performed for the WHDF elliptical galaxies.
For field objects with z > 0.25 a transforma-tion to B rest–frame was conducted based on the B passband and for 0.55 ≤ z < 0.75 the RC filter was utilised. Nevertheless, for internal consistency the rest-frame magnitudes derived with the FORS filter systems were checked by transforming the observed F814W-magnitudes
into rest-frame Johnson-B. The deviations be-tween magnitudes were small (differences were less than their errors).
For testing purposes, the colors of the templates were derived purely within the Johnson-Cousins Filter system and afterwards compared to the values given by Fukugita et al. The deviations were very small, with absolute colors (X −Y) within the range 0.03m ≤ |∆(X−Y)| ≤0.08m.