Annu. Rev. Astron. Astrophys. 1988. 26: 561-630 Copyright © 1988 by Annual Reviews. All rights reserved

4.2. Conversion of Observed Heterochromatic Apparent Magnitude to the Apparent Bolometric Scale: The K Correction

The shift of the spectrum toward the red causes observed apparent magnitudes to differ from those that would have been observed at zero redshift. The correction is due to the fixed detector effective wavelength and the finite detector bandwidth. As defined in current usage (HMS, appendix B; Oke & Sandage 1968), it is composed of two terms. The effective bandwidth in the rest frame of the source is smaller than in the rest frame of the observer because the source spectrum is stretched upon redshift. Each rest-frame wavelength ₀ appears to the observer at ₀(1 + z), whereas the detector bandwidth ₀ is generally fixed. The bandwidth term due to this stretching is 2.5 log(1 + z) magnitudes, in the sense that the corrected magnitude must be brighter than the observed. The color-selective term is the ratio of the flux of the redshifted spectrum to the unshifted spectrum that is accepted by the detector. The term can be calculated by quadrature once the rest-frame spectral energy distribution (SED) of the source is known (the second term in Equation B7 of HMS).

Lack of accurate knowledge of the K correction was a stumbling block in the early interpretations of (a) the galaxy counts (Hubble 1936b, Greenstein 1938), (b) the m(z) magnitude-redshift relation (Hubble 1953, HMS), and (c) the color evolution (Stebbins & Whitford 1948). Because of its crucial role in the interpretation of these cosmological test data, great effort was made from 1960 to 1975 to measure the SED of galaxies of various Hubble types. Early emphasis was put on E and S0 galaxies because of their dominance as first-ranked cluster galaxies. However, for the galaxy count problem in the field, K(z) values for spirals of all Hubble classes (Sa, Sb, Sc, Sd, Sm, Im) are also required, together with knowledge of the fractional morphological mix of the sample.

Following Stebbins & Whitford's (1948) early six-color broadband measurements that gave highly smoothed I() distributions (see also Whitford 1954), Code (1959) and Oke & Sandage (1968) obtained spectrum scanner data at 50 and 25 Å resolution of bright E galaxies. A study using intermediate-band photometry was also made by Lasker (1970). These gave the first modern K corrections during the 1970s, although the data referred only to the central ~ 15-arcsec regions of E galaxies in the Leo and Virgo clusters. Because the centers of E and S0 galaxies are redder than the outer regions (de Vaucouleurs 1960, Tifft 1963, 1969, de Vaucouleurs & de Vaucouleurs 1972, Sandage & Visvanathan 1978), these nuclear K(z) values were too large by progressive factors that reach ~ 0.1 mag at z ~ 0.3 (Whitford 1971, his Figure 2). Schild & Oke (1971) and Whitford (1971) then used very large aperture photometry to account for the color gradient. From their integrated SEDs they calculated K(z) values in the B, V, and R photometric bands to redshifts of z = 0.28 for B and of z = 0.60 for V and R, but again only for E and S0 galaxies. Oke (1971) then obtained spectral scans of three distant (at that time) first-ranked cluster galaxies at redshifts of z = 0.2, z = 0.38, and z = 0.46, giving usable SEDs for E galaxies to ₀ = 2700Å in the rest frame. This permitted K(B) to be calculated to z = 0.52 and K(V) and K(R) to z = 0.72 on the assumption of no color evolution.

Wells (1972) measured I() from 3500 to 5500 Å for a range of galaxy types. Using these data, to which OAO-2 data (Code et al. 1972) were added in the near-UV, Pence (1976) calculated K corrections for all galaxy types to large redshifts, giving very useful comprehensive tables. Further UV data were added by Ellis et al. (1977). Using the final reduced data from the OAO-2, Code & Welch (1979) calculated new K corrections to redshifts of z = 1. Using these data, together with new observations made with the ANS satellite, Coleman et al. (1980) calculated I() energy distributions for old stellar populations and for Sbc, Scd, and Im galaxies to ₀ = 1400Å and gave comprehensive K corrections in the U, B, V, and R photometric bands to z = 2. These data are the most extensive K corrections now available in the standard broadband photometric system, providing an enormous advance in this crucial problem since the early analysis by HMS. Sebok (1986), using the energy distributions of Wells, listed K(z) for all morphological types for the Thuan-Gunn red system. Schneider et al. (1983a) list K(z) for the Thuan-Gunn g, r, i, and z bands for giant E and S0 galaxies.

A summary of the SEDs in the archive literature that have been used to calculate Hand the color variations with z is given by Yoshii & Takahara (1988) in their valuable review of cosmological tests.