Next Contents Previous


Consider a source observed to have apparent magnitude mR when observed through photometric bandpass R, for which one wishes to know its absolute magnitude MQ in emitted-frame bandpass Q. The K correction KQR for this source is defined by

Equation 2 (2)

where DM is the distance modulus, defined by

Equation 3 (3)

where DL is the luminosity distance (e.g. Hogg 1999) and 1 pc = 3.086 × 1016 m.

The apparent magnitude mR of the source is related to its spectral density of flux fnu(nu) (energy per unit time per unit area per unit frequency) by

Equation 4 (4)

where the integrals are over the observed frequencies nu0; gRnu(nu) is the spectral density of flux for the zero-magnitude or "standard" source, which, for Vega-relative magnitudes, is Vega (or perhaps a weighted sum of a certain set of A0 stars), and, for AB magnitudes (Oke & Gunn 1983), is a hypothetical constant source with gABnu(nu) = 3631 Jy (where 1 Jy = 10-26 W m-2 Hz-1 = 10-23 erg cm-2 s-1 Hz-1) at all frequencies nu; and R(nu) describes the bandpass, as follows:

The value of R(nu) at each freqency nu is the mean contribution of a photon of frequency nu to the output signal from the detector. If the detector is a photon counter, like a CCD, then R(nu) is just the probability that a photon of frequency nu0 gets counted. If the detector is a bolometer or calorimeter, then R(nu) is the energy deposition h nu per photon times the fraction of photons of energy nu that get absorbed into the detector. If R(nu) has been properly computed, there is no need to write different integrals for photon counters and bolometers. Note that there is an implicit assumption here that detector nonlinearities have been corrected.

The absolute magnitude MQ is defined to be the apparent magnitude that the source would have if it were 10 pc away, at rest (i.e., not redshifted), and compact. It is related to the spectral density of the luminosity Lnu(nu) (energy per unit time per unit frequency) of the source by

Equation 5 (5)

where the integrals are over emitted (i.e., rest-frame) frequencies nue, DL is the luminosity distance, and Q(nu) is the equivalent of R(nu) but for the bandpass Q. As mentioned above, this does not require Q = R, so this will lead to, technically, a generalization of the K correction. In addition, since the Q and R bands can be zero-pointed to different standard sources (e.g., if R is Vega-relative and Q is AB), it is not necessary that gQnu = gRnu.

If the source is at redshift z, then its luminosity is related to its flux by

Equation 6 (6)

Equation 7 (7)

The factor of (1 + z) in the luminosity expression (6) accounts for the fact that the flux and luminosity are not bolometric but densities per unit freqency. The factor would appear in the numerator if the expression related flux and luminosity densities per unit wavelength.

Equation (2) holds if the K correction KQR is

Equation 8 (8)

Equation (8) can be taken to be an operational definition, therefore, of the K correction, from observations through bandpass R of a source whose absolute magnitude MQ through bandpass Q is desired. Note that if the R and Q have different zero-point definitions, the gRnu(nue) in the numerator will be a different function from the gQnu(nu0) in the denominator.

In equation (8), the K correction was defined in terms of the apparent flux fnu(nu) in the observed frame. This is the direct observable. Most past discussions of the K correction (e.g. Oke & Sandage 1968; Kim et al. 1996) write equations for the K correction in terms of either the flux or luminosity in the emitted frame. Transformation from observed-frame flux fnu(nu0) to emitted-frame luminosity Lnu(nue) gives

Equation 9 (9)

In the above, all calculations were performed in frequency units. In wavelength units, the spectral density of flux fnu(nu) per unit frequency is replaced with the spectral density of flux flambda(lambda) per unit wavelength using

Equation 10 (10)

Equation 11 (11)

where c is the speed of light. The K correction becomes

Equation 12 (12)

where, again, R(lambda) is defined to be the mean contribution to the detector signal in the R bandpass for a photon of wavelength lambda and Q(lambda) is defined similarly. Note that the hypothetical standard source for the AB magnitude system, with gABnu(nu) constant, has gABlambda(lambda) not constant but rather gABlambda(lambda) = c lambda-2 gABnu(nu).

Again, transformation from observed-frame flux flambda(lambdao) to emitted-frame luminosity Llambda(lambdae) gives

Equation 13 (13)

Equation (13) becomes identical to the equation for K in Oke & Sandage (1968) if it is assumed that Q = R, that gQnu = gRnu, that the variables lambda0, F(lambda), and Si(lambda) in Oke & Sandage (1968) are set to

Equation 14 (14)

and that the integrand lambda is used differently in each of the two integrals. Similar transformations make the equations here consistent with those of Kim et al. (1996), although they distinguish between the classical K correction and one computed for photon counting devices (an unnecessary distinction); their most similar equation is that given for Kxycounts.

Next Contents Previous