Calculating restframe magnitudes and colours for galaxies and QSOs

The multi-colour classification performed in the COMBO-17 and MANOS surveys produces estimates for the redshift and spectral type of galaxies and QSOs which are best fits to the measured multi-band SED. Given the spectral coverage from U- to I-band, it is possible to estimate restframe luminosities within this wavelength range directly from the measured photometry. If a restframe band falls outside the U-to-I range because of the redshift, then estimating the luminosity involves an estimated extrapolation of the SED beyond the range probed by the photometry. This is called K-correction.

The determination of luminosities involves two parts, a cosmological part unique to the redshift value, and a spectrophotometric part unique to the object. The former consists of calculating a distance modulus based on a set of cosmological parameters. In COMBO-17, we have so far published values for (Omega-m,Omega-l,H0)=(0.3,0.7,100). Changing the Hubble constant means only shifting the zeropoint of the distance modulus scale, e.g. H0=70 means that all distance moduli have to be increased by -0.775 mag. In contrast, changing the cosmological parameters means that the distance moduli change in a non-trivial redshift-dependent way. The prgs for restframe calculation in COMBO-17 also calculate luminosity distances and luminosities for (Omega-m,Omega-l,H0)=(1.0,0.0,100) and (Omega-m,Omega-l,H0)=(0.2,0.0,100).

The spectrophotometric part unique to every object places the best-fitting template into the observed SED by linking it to the measured broadband nearest to the given redshifted restframe band in question (with a bias towards the red, where we expect higher and more reliably measured flux from the object). Here, a K correction table is used which in design looks entirely like a colour library and contains all galaxy (or QSO) templates for the necessary range of redshifts as one object per row with the colour indices in the columns. However, the colour indices in the K correction table are not *observed-frame*-minus-*observed-frame* like in the colour library used for the classification; instead they are *restframe*-minus-*observed-frame*, linking various interesting restframe bands (Jonhson, Bessell, SDSS, ...) to the various observed-frame ones that appear in the survey data (WFI-UBVRI).

The practical procedure to calculate restframe luminosities is then as follows:

As a result you get luminosities and restframe colours. The errors for the luminosities take into account a constant minimum error for absolute calibration of 0.1 mag, the error of mag_best which defines the total magnitude and hence the aperture correction, and the error of the nearest observed-frame band which drives the normalisation of the template (all added in quadrature). The errors of the restframe colours are just the quadratically added errors from the two observed-frame bands near the two restframe bands combined in the colour index and a 0.1 mag error floor added in again. But the aperture correction to total magnitudes is not used as it does not affect the colour in the absence of colour gradients. With colour gradients being present, it is not clear how to estimate the error from the existing data easily, and we hope that the effect is accounted for by the error floor. Also, the error in mag_best is not included as it does not affect the colours either.



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Last update July 27, 2005, CW