## 7. Problems of slope and calibrationIn the discussion thus far, we have assumed that the normalized
distance has been accurately calculated, so that there are no
observational errors in and the
kinematical distance scale () is
accurate as well. However, this cannot be the case, and as a result,
field galaxies collected from a fixed true distance will show in the
Two questions arise: first, in such a situation, what happens to the slope-criterion requiring that for the unbiased plateau galaxies goes horizontally in the vs. diagram? Secondly, how do we calculate the actual value of with calibrators known to have a different iTF slope? Here we expand somewhat the compact treatment in Teerikorpi (1990). Consider two galaxy clusters in the Assume now that the calibrators follow the slope . One may describe in a picturesque manner how to do the comparison with calibrators, leading to an unbiased value of . As the shallower slope for the field galaxies is assumed to be due to errors in , one may think of "adding" random errors to calibrator linear log diameters , until the slope falls to and then use the obtained relation for calibration ("shifting of regression lines"). Fortunately it is not necessary to actually "shuffle" 's - this is equivalent to forcing the slope through the calibrators, at least in an ideal situation. So, let us assume first that the calibrator sample is volume-limited. Then adding random errors to means that remains the same, and one arrives at the relation On the other hand, the actual calibrators follow Hence, after this trick, the new zero-point which must be applied for the sample galaxies is: There is also a longer route to this result, using Eq. (7) giving the direct relation in terms of the inverse parameters and . If one increases so that and requires that the direct relation does not change, one also arrives at Eq. (24). On the other hand, the formula for is exactly what is obtained if one simply forces the slope through the calibrators. From this follows that a correct calibration is achieved by deriving the zero-point by a force-fit of the slope through the calibrators. © European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |