9. Another route: using calibrator slope and making and Malmquist corrections
In principle, the problem of the p-distribution of the calibrators does not affect the determination of the zero-point when one forces the correct slope on the calibrators. This makes one ask what happens if and are then used for the field sample. First of all one expects a distance-dependent bias, according to Eq. 15 from Teerikorpi (1990), here adopted to diameters:
Here is the average (log) diameter for the galaxies at true distance r. Now one may ask what is the average bias for the sample? The average bias clearly depends on the average of . This is actually the Malmquist bias of the 1st kind for Gaussian distribution of diameters with dispersion , and in the case of a space density distribution one gets:
When the distribution is homogeneous. In order to use this as a correction, one must know both and . Clearly, if the field and calibrator slopes are identical, then is unbiased, and this is so irrespective of the nature of the calibrator sample. Note also that this formula contains the width of the total diameter function, not the smaller dispersion . Hence the effect can be formidable even when the slopes and do not differ very much.
As a concrete example, consider the KLUN field galaxies following the inverse slope which is in agreement with the simple disk model of Theureau et al. (1997a). This slope, when forced through the calibrators, gives . Above it was noted that the calibrator sample cannot represent the cosmic distribution of , and this is the situation when a too high is derived if the actual slope is steeper than the slope from the field galaxies. It is natural to ask, how large a slope would in this manner explain the difference between and . Taking , we get predicted for different slopes as given in Table 2.
Table 2. as derived using when and the true calibrator slope is .
The last column corresponds to a decrease in the average space density, corresponding to density as derived by Teerikorpi et al. (1998). It may be concluded that in order to explain the difference between and 55 solely in this manner, it is required that actually 0.65-0.75, instead of . Ekholm et al. (1999) show that such a steep calibrator slope really is consistent with the data.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999