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
where DM is the distance modulus, defined by
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 f() (energy per unit time per unit area per unit frequency) by
where the integrals are over the observed frequencies 0; gR() 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 gAB() = 3631 Jy (where 1 Jy = 10-26 W m-2 Hz-1 = 10-23 erg cm-2 s-1 Hz-1) at all frequencies ; and R() describes the bandpass, as follows:
The value of R() at each freqency is the mean contribution of a photon of frequency to the output signal from the detector. If the detector is a photon counter, like a CCD, then R() is just the probability that a photon of frequency 0 gets counted. If the detector is a bolometer or calorimeter, then R() is the energy deposition h per photon times the fraction of photons of energy that get absorbed into the detector. If R() 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 L() (energy per unit time per unit frequency) of the source by
where the integrals are over emitted (i.e., rest-frame) frequencies e, DL is the luminosity distance, and Q() is the equivalent of R() 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 gQ = gR.
If the source is at redshift z, then its luminosity is related to its flux by
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) 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 gR(e) in the numerator will be a different function from the gQ(0) in the denominator.
In equation (8), the K correction was defined in terms of the apparent flux f() 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 f(0) to emitted-frame luminosity L(e) gives
In the above, all calculations were performed in frequency units. In wavelength units, the spectral density of flux f() per unit frequency is replaced with the spectral density of flux f() per unit wavelength using
where c is the speed of light. The K correction becomes
where, again, R() is defined to be the mean contribution to the detector signal in the R bandpass for a photon of wavelength and Q() is defined similarly. Note that the hypothetical standard source for the AB magnitude system, with gAB() constant, has gAB() not constant but rather gAB() = c -2 gAB().
Again, transformation from observed-frame flux f(o) to emitted-frame luminosity L(e) gives
Equation (13) becomes identical to the equation for K in Oke & Sandage (1968) if it is assumed that Q = R, that gQ = gR, that the variables 0, F(), and Si() in Oke & Sandage (1968) are set to
and that the integrand 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.