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B. Correlation between tex2html_wrap_inline3443 and tex2html_wrap_inline3445

Hereafter, the correlation between p and the velocity estimator tex2html_wrap_inline2815 given Eq. (20 (click here)) is calculated assuming the following hypothesis:

Under assumptions tex2html_wrap_inline27451, tex2html_wrap_inline27452 and tex2html_wrap_inline27453, the probability density tex2html_wrap_inline3469 of Eq. (11 (click here)) rewrites:
In order to calculate each term involved in the covariance of p and tex2html_wrap_inline2815, i.e. tex2html_wrap_inline3475, the velocity estimator tex2html_wrap_inline2815 given Eq. (9 (click here)) is rewritten in terms of variables v, tex2html_wrap_inline2707 and tex2html_wrap_inline2727:
Since u is a constant, tex2html_wrap_inline3487. Integrating over the variable v gives for the three quantities tex2html_wrap_inline3491, tex2html_wrap_inline3493 and tex2html_wrap_inline3495:


where properties (A5 (click here)a) and (A5 (click here)b) have been used. Replacing tex2html_wrap_inline2739 by tex2html_wrap_inline3499 and using definition of B given Eq. (9 (click here)), Eqs. (B3 (click here), B4 (click here), B5 (click here)) expressed in terms of p, m and tex2html_wrap_inline2707 read:


where the constant C is defined as follows:
By using properties (A4 (click here)) and (A5 (click here)a), it is found that integral of Eq. (B9 (click here)) vanishes. It thus implies that C=0, tex2html_wrap_inline3513, tex2html_wrap_inline3515, tex2html_wrap_inline3517 and finally tex2html_wrap_inline3519. This result does not depend on the shape of the selection function tex2html_wrap_inline2733 in m and p, on the theoretical distribution function fp(p) of the variable p and on the mean radial peculiar velocity u along the line-of-sight. For a sample of galaxies satisfying hypotheses tex2html_wrap_inline27451, tex2html_wrap_inline27452 and tex2html_wrap_inline27453, it thus turns out that the observable p is not correlated with the velocity estimator tex2html_wrap_inline2815 as long as the model parameters entering into tex2html_wrap_inline3543 via Eq. (9 (click here)) are correctgif. On the other hand, if a wrong value is assumed for one of these calibration parameters, say for example that aD, bD and tex2html_wrap_inline2719 are correct but not the Hubble constant tex2html_wrap_inline3559, the covariance between p and tex2html_wrap_inline3563 gives:
which does not vanish since existing selection effects on apparent magnitude m correlate variables p and tex2html_wrap_inline2761 (i.e. for selected galaxies, observable p increases in average with the distance estimator tex2html_wrap_inline2761). This point motivates the use of a null-correlation approachgif in order to calibrate Tully-Fisher like relations.

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