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Astron. Astrophys. 343, 713-719 (1999)

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7. Problems of slope and calibration

In the discussion thus far, we have assumed that the normalized distance has been accurately calculated, so that there are no observational errors in [FORMULA] and the kinematical distance scale ([FORMULA]) 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 p vs. [FORMULA] diagram a different slope than without such errors. This slope will differ from the intrinsic slope followed by calibrators.

Two questions arise: first, in such a situation, what happens to the slope-criterion requiring that [FORMULA] for the unbiased plateau galaxies goes horizontally in the [FORMULA] vs. [FORMULA] diagram? Secondly, how do we calculate the actual value of [FORMULA] 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 p vs. [FORMULA] diagram, both having (in their unbiased [FORMULA]-part), the same iTF slope [FORMULA], which however differs from the calibrator slope. It is evident that the slope [FORMULA] gives the correct relative distance of these clusters, when one shifts the regression lines one over the other (c.f. Fig. 1). Hence, as an answer to the first question, this is also the slope which corresponds to the horizontal distribution of [FORMULA]; correct distance ratios reveal the linear Hubble law.

Assume now that the calibrators follow the slope [FORMULA]. One may describe in a picturesque manner how to do the comparison with calibrators, leading to an unbiased value of [FORMULA]. As the shallower slope for the field galaxies is assumed to be due to errors in [FORMULA], one may think of "adding" random errors to calibrator linear log diameters [FORMULA], until the slope falls to [FORMULA] and then use the obtained relation for calibration ("shifting of regression lines"). Fortunately it is not necessary to actually "shuffle" [FORMULA]'s - this is equivalent to forcing the slope [FORMULA] 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 [FORMULA] means that [FORMULA] remains the same, and one arrives at the relation

[EQUATION]

On the other hand, the actual calibrators follow

[EQUATION]

Hence, after this trick, the new zero-point which must be applied for the sample galaxies is:

[EQUATION]

There is also a longer route to this result, using Eq. (7) giving the direct relation in terms of the inverse parameters and [FORMULA]. If one increases [FORMULA] so that [FORMULA] and requires that the direct relation does not change, one also arrives at Eq. (24).

On the other hand, the formula for [FORMULA] is exactly what is obtained if one simply forces the slope [FORMULA] through the calibrators. From this follows that a correct calibration is achieved by deriving the zero-point by a force-fit of the slope [FORMULA] through the calibrators.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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