Astron. Astrophys. 339, 647-657 (1998)

5. Including K-effect and other corrections

A realistic situation involves magnitudes defined by a fixed wavelength band, which in the presence of redshift requires one to take into account the K-effect. We do not see in the sky the bolometric magnitude , but

In the calculation using bolometric magnitude, the distribution of at m was derived from the knowledge of the luminosity function where . With the K-correction, one instead has to use in order to get the absolute magnitude that defines the luminosity function.

Generally, if increases with z so that the galaxies get faint more rapidly than just due to the bolometric factor, then the backside volume effectively decreases in Friedmann models, and the trend noted with bolometric magnitudes is still strengthened. At a fixed observed the absolute magnitude is reached at a smaller z than if the K-correction would be zero ().

5.1. "Elliptical" -corrections

We illustrate this effect using the K-correction for the V-magnitudes of elliptical galaxies. Coleman et al. (1980) give such corrections up to in their Table 7, corresponding to measurements of the bulge of M31. Without evolution, the K-effect would make an elliptical galaxy fainter by 3.46 and 5.07 mag at 1.0 and 2.0, respectively, as compared with the pure effect of the bolometric factor. As we make calculation only up to , it is sufficient to extrapolate the K-corrections up to .

Table 1 shows the ratios of foreground and background volumes in this case for . Note how the ratios change in comparison to the bolometric case. The change of the trend around is caused by the fact that there the change of bolometric magnitude with z becomes more rapid than the change in the K-correction.

It is a practice to include the K-correction in the theoretical zero-dispersion Mattig curve and compare this with data in the Hubble diagram (e.g. Yoshii & Takahara 1988). This means that the Mattig equation becomes

from which the relation is solved. Now the relevant question is: How much do the actual vs. relation, derived as discussed above, and this K-shifted zero-dispersion curve differ from each other?

Fig. 6 shows similarly as Figs. 4 and 5 the difference between the actual and the predicted by the K-shifted Mattig vs. m relation. In comparison with the bolometric magnitudes, the "apparent" inferred using the K-shifted curves becomes still smaller, and the effect may be quite significant even for good standard candles. For example, if is actually 1.0 and mag, the usual procedure would make one infer that . If is 0.5, then one might derive that . With a larger , the apparent values of would become still lower (e.g. with mag, would drop down to ).

Fig. 6a and b. The difference between and the zero-dispersion, K-corrected (Mattig) prediction for a standard candle class with and mag and "elliptical" K-correction for V-magnitudes (dots). The Friedmann universe has km s-1 Mpc-1 and (upper panel) or = 1.0 (lower panel). The open triangles show the corresponding difference for an apparent value of which roughly predicts the observed at brighter magnitudes. Around , or .

5.2. Why not make K-corrections to observed magnitudes and then compare with "bolometric" predictions?

At this point it is useful to consider the question: What if one first constructs the vs. diagram where and then compares the resulting run of data points with the more fundamental bolometric predictions of Sect. 4? Using this, perhaps "more natural" procedure, does one generally get the same result on , as with the vs. diagram? This question is relevant, because if the answer is positive, it is sufficient to work with the results discussed for bolometric magnitudes.

However, the answer is negative, which may be seen as follows: The procedure with bolometric magnitudes requires that the distribution of at constant m does not depend on anything else than space geometry, bolometric factor , and luminosity function. A constant is made from galaxies with different and the distribution of at now comes to depend also on the relative completeness of the sample at different . For example, if the K-correction increases towards high z, the high-z wing of the -distribution at a constant will be underpopulated relative to the small-z wing. This is schematically illustrated in Fig. 7, where are shown 1) a distribution of at a constant bolometric magnitude (open circles), 2) positions of these galaxies in the Hubble diagram influenced by the K effect (dots), together with a vertical line indicating the magnitude limit, and 3) the galaxies after K-correction made for the observed galaxies (triangles). Clearly, making K-corrections does not reproduce the bolometric distribution, because many high-z galaxies have been lost beyond the magnitude limit. By the way, here we have a situation where the two types of biases together influence interpretation of the Hubble diagram even in the sense of vs. m.

Fig. 7. Part of a schematic Hubble diagram showing how making negative K-corrections to observed magnitudes (dots) affected by a magnitude limit (here = 24 mag) does not reproduce the distribution of for bolometric magnitudes (open circles), but a truncated distribution results (triangles). In this case the selection bias influences the cosmological Malmquist bias. This implies that one should use in the analysis of the Hubble diagram versus (and the results of Sect. 5.2), and not versus K-corrected (with the bolometric results of Sect. 4), especially close to the magnitude limit.

It should be added that a similar problem appears also if the K-corrections have not a positive, but negative sign (like typically radio loud quasars in V-band; e.g. Fig. A1 in Teerikorpi 1981). In this case, a lower part of the distribution at faint apparent magnitudes corrected for the K-effect is missing, because of the magnitude limit.

In the approach of Sect. 5.1. the magnitude incompleteness does not appear for the same reason as in the formalism using bolometric magnitude (in fact, instead of , one has , and galaxies with apparent magnitude m are gathered unaware of whether the faintness is due to the cosmological bolometric factor f or the K-factor K).

It may be concluded that the vs. approach is really the recommended one, and the theoretical expectations must be calculated in the manner described in Sect. 5.1. Of course, when one works sufficiently far away from the magnitude limit, then the K-corrected magnitudes , together with the bolometric predictions (Sect. 4), should give the same estimate of .

5.3. Luminosity evolution, galactic and intergalactic extinction

The present paper deals with the fundamental modifications that the classical Malmquist bias experiences, when standard candles are put into a non-classical, here Friedmann, space from small to high redshifts. In this first step, a few basic assumptions are still taken over from the classical case: 1) the standard candle class itself and its space number density do not change with the look-back time, 2) the intergalactic space is clean from extinction, and 3) there is also no variable foreground galactic extinction screen. Here we make just a few comments on these topics.

As far as the observer is concerned, a change of the intrinsic luminosity with redshift has similar consequencies for the vs. diagram as a K-effect, and must be included in the analysis formally as was done in Sect. 5.1. The influence of intergalactic extinction is like that of K-correction and z-dependent evolution. Though it is not presently known how important this extinction is, it would in any case work in the same direction as an increasing K-term, i.e. pulling down the apparent value of .

How should one make galactic extinction corrections? Consider two regions of the sky, one (A) without extinction and another (B) with a constant extinction . At A, the galaxies falling on the apparent magnitude m have the same distribution of , as the galaxies at B, found at the apparent magnitude . Hence, one may freely combine in the vs. m diagram the galaxies at A and at B, just by using for the latter the corrected . In this manner, the vs. m relation is not affected by galactic extinction, if each m is accurately corrected (so, the procedure differs from K-, evolution, and intergalactic extinction corrections).

© European Southern Observatory (ESO) 1998

Online publication: October 22, 1998
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