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Astron. Astrophys. 362, 851-864 (2000) 4. An application: the FP of cluster galaxiesThe use of the FP as distance indicator is based on the hypothesis that its slopes and thickness are `universal', i.e., independent of the sample of galaxies. The problem of the universality has been studied by JFK96 by deriving the FPs of ten clusters of galaxies. Although the authors find that the FP coefficients are no significantly different from cluster to cluster, they admit that the samples are too small to get an accurate comparison. Up to now, the FP has been derived with a significant number of
galaxies only for Coma cluster ( On the basis of the discussion in Sects. 2 and 3, we try now to address the question of the universality by comparing the FPs of different clusters. We chose the samples so that a) their size was as large as
possible; b) the FP parameters of the different samples were
homogeneous; c) each sample had photometric parameters in the same
waveband and d) the cluster were at Table 2. Column 1: cluster identification. Column 2: CMB-frame redshift. Column 3: limiting apparent magnitudes. The values were referred to the distance of Coma by using the redshifts of Column 2. Column 4: number of galaxies (E+S0). Column 5: references from which the FP parameters are drawn. Columns 6, 7: FP slopes, a and b, obtained by the The FP parameters, The FP coefficients were determined by the
To compare the FP slopes, it should be taken into account that the
uncertainties on a and b can be correlated. By
Eqs. B17, we obtained the theoretical estimates of the CM
components of the slopes. These estimates were then used to derive the
ellipse that defines a
The Fig. 10 has the disadvantage to be not easily readable. To
obtain a more immediate description we compared separately the
a and b values as derived from different fits, as
plotted in Fig. 11. The FP slopes of the
All the samples (except one) have small size,
Although the coefficients a and b are consistent for
each pair of samples in every fitting procedure, we see that the error
bars are very large, typically It is also worth to notice that in the
To address this point, we tried to correct the FP slopes for the different magnitude-completeness of the samples (see Giovanelli et al. 1997 and Scodeggio et al. 1998 for a wide discussion). At first, we constructed the completeness histograms of each sample. To this aim, the magnitude range was binned and the fraction of galaxies of each bin normalised to the corresponding fraction of the Coma photometric sample of JFK95a. This data set is complete in fact out to a magnitude higher than the magnitude limits of the other samples (see Table 2 and references therein). It turns out that the same results are also obtained by normalizing the fractions of galaxies through a gaussian model of the luminosity function (see Scodeggio et al. 1998). The histograms were then modeled by Fermi-Dirac distributions and incomplete FP simulations constructed through the modeled distributions. By fitting the simulations, we estimated the corrections on a and b. As shown in Fig. 11, the corrections shift upwards the
coefficient a and reduce the correlation with the sample size
and the systematic difference of the Coma sample. It turns out, in
fact, that these effects are a consequence of the different
completeness of the samples with respect to the photometric
parameters. This is shown by the results of the `inverse' fit, with
It is also evident that a systematic difference is present between the FP coefficients derived from the various MIST fits. As discussed in Sect. 2.2, the `fitting bias' is due to the lack of a model for the intrinsic scatter of the FP. For the same reason, only the projection of the intrinsic dispersion along some assigned direction, with respect to a given fitted plane, can be measured. For instance, we calculate the scatter projection on the
The In Table 3, we show the weighted means of the coefficients
a and b for the various fitting methods. The means were
calculated after the magnitude-completeness and the
Table 3. Weighted means of the `corrected' FP slopes (see text). Column 1: fitting method. Columns 2 and 3: mean values of a and b with corresponding uncertainties (1 The difference between the various MIST fits amounts to
The bias that could be introduced by neglecting the measurement
errors was found to be negligible for b
( For what concerns the covariance term of the photometric
parameters, as discussed in Sect. 2.2, it was estimated by the
relation The systematic dependence of the FP coefficients from the fitting method can also be seen by comparing the values obtained by the MIST fits with the results of JFK96 and HLS97. In Fig. 12 the MIST estimates of a and b are
compared to the values of JFK96, that adopted an ORLS (robust) method.
To this aim, we plot the differences of the FP slopes (JFK96 - ours)
divided by the relative
The values of b agree with those by JFK96 for the
The values of the FP slopes obtained by JFK96 for the whole cluster
sample, HLS97 derive the FP slopes for a sample of seven clusters of
galaxies. By adopting the The slopes of the individual clusters of HLS97 are derived by a
bi-dimensional fit, adopting as independent variable the combination
of Since the ORLS and OLS methods do not account for the measurement errors, and due to differences in completeness and selection, a detailed explanation of the origin of the above discrepancies is not achievable. ![]() ![]() ![]() ![]() © European Southern Observatory (ESO) 2000 Online publication: October 30, 2000 ![]() |