K-correction Basics

The basics of K-corrections are well-described in Hogg et al. (2002) and in Blanton & Roweis (2007). Here is a briefer description including how they are estimated by kcorrect.

If you want to convert apparent magnitudes in band R to absolute magnitudes in band Q, you need to calculate the K-correction, which is defined by the equation:

\[m_R = M_Q + {\rm DM}(z) + K_{QR}(z),\]

where \(m_R\) is the apparent magnitude, \(M_Q\) is the absolute magnitude, \({\rm DM}(z)\) is the distance modulus, accounting for the angular diameter distance and cosmological surface-brightness dimming, and \(K_{QR}(z)\) is the K-correction.

By absolute magnitude we mean: the apparent magnitude in band \(Q\) that the object would have if it were observed at rest, 10 pc away, using an aperture that contains its total flux. The distance modulus accounts for the difference between an object’s actual distance and 10 pc. The K-correction accounts for the fact that you observed a redshifted galaxy in band \(R\) but the absolute magnitude requires a rest-frame observation in band \(Q\). Obviously the difference between the fluxes observed in different bandpasses is fully determined by the galaxy SED and the description of the bandpasses.

In order to get the appropriate SED for a set of galaxy fluxes, kcorrect fits an SED which is a nonnegative linear combination of some small number of templates. The templates have been optimized to minimize the residuals between the actual galaxy fluxes and the galaxy fluxes reconstructed from the galaxy SED fit. The K-correction is then calculated from this best-fit SED.

To perform the fits, the software requires broad band flux measurements. These are accepted as AB maggies. Maggies are the ratio of the source to the AB standard source in each band, using the integrals in the kcorrect paper. They have a simple relationship to magnitudes:

\[m = − 2.5 \log 10 \mu.\]

where \(m\) is the magnitude and \(\mu\) is maggies. An advantage of the maggie unit system relative to magnitudes is that it is linear, and thus can when necessary accommodate negative flux estimates.

The AB standard source is a flat spectrum object with \(f_\nu = 3631 {\rm ~Jy} = 3.631 \times 10^{−20} {\rm ~ergs} {\rm ~cm}^{−2} {\rm ~s}^{−1} {\rm ~Hz}^{−1}\). Such a source would have all magnitudes equal to zero.

There are still many available catalogs that are defined on the Vega standard source system. The Response object associated with a filter has the vega2ab attribute which defines the correction from Vega to AB in magnitudes.

The band \(Q\) can in principle be anything; it does not have to be the same as \(R\), and it also does not have to be an actual bandpass at all.

kcorrect supports a particular choice of a “shifted” bandpass with its band_shift option, where a band shift of \(z\) would be denoted \(^{z}Q\) and would indicate that the rest-frame band pass \(Q\) blue shifted by a factor \((1+z)\). In this case the K-correction for an object at redshift \(z\) from observed bandpass \(R\) to a rest frame band pass \(^{z}R\) would be independent of the object’s SED and equal to \(-2.5\log_{10}(1+z)\), because the bandpasses exactly overlap. The advantage of this choice is that by choosing the band shift to be near the typical redshift of a sample, one can minimize the errors due to K-corrections when comparing objects within the sample.

Two final notes on SDSS units, which since kcorrect grew out of SDSS work seems appropriate here.

  • The official SDSS catalog numbers are not our best guess for the AB system, so SDSS data has a small conversion factor that needs to be applied using sdss_ab_correct().

  • The “magnitudes” published by SDSS are so-called “asinh magnitudes” or “luptitudes”, described by the SDSS imaging documentation. These can be converted to maggies with sdss_asinh_to_maggies().