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

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4. Analytic calculation of (log H) vs. da

Assuming Gaussian distributions and sharp cutoffs at [FORMULA] and [FORMULA], it is easy to calculate the expected run of [FORMULA] vs. normalized distance [FORMULA]. For simplicity, we write the inverse relation as

[EQUATION]

We use [FORMULA] as shorthand notation for the logarithm of the linear diameter. At any fixed [FORMULA] the dispersion of p is [FORMULA].

Now the task is to calculate the bias in [FORMULA] at each [FORMULA]. This is done starting from the same initial formula (28) of T84 which was used to show the absence of bias. However, in this case one must add the selection function [FORMULA]:

[EQUATION]

where [FORMULA] and [FORMULA].

This formula gives for the sample galaxies with true (log) diameter [FORMULA], the (log) error in the diameter deduced by the inverse relation (using [FORMULA]). It is directly reflected to the derived value of [FORMULA] at constant [FORMULA] (i.e., at fixed normalized distance [FORMULA]).

With the present choice of the selection function (sharp upper and lower cutoffs), and after a change of variable,

[EQUATION]

Eq. (4) becomes

[EQUATION]

Here the term [FORMULA] can be evaluated in terms of the error function [FORMULA]:

[EQUATION]

where [FORMULA] and [FORMULA] correspond to the cutoffs [FORMULA] and [FORMULA] via Eq. (5).

Inspection of the formula confirms the qualitative conclusions of Sect. 2. Putting [FORMULA], so that only the upper cutoff remains (and recalling that [FORMULA]), the bias is seen to be positive: the diameters are underestimated, leading to underestimated distances, hence to overestimated [FORMULA]. Note that this calculation does not tell us the average bias in the sample.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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