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Astron. Astrophys. 362, 851-864 (2000)

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4. An application: the FP of cluster galaxies

The 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 ([FORMULA]). Most of the other cluster samples have [FORMULA] (e.g. Dressler et al. 1987, JFK96, Hudson et al. 1997, Pahre et al. 1998).

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 [FORMULA] 0. The four prescriptions are needed to a), b) make significant comparisons, c) avoid waveband dependencies, and d) avoid galaxy evolution effects. The ten samples surviving the above criteria are listed in Table 2.


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 [FORMULA] fit, with [FORMULA] standard intervals. Columns 8, 9: FP slopes and relative uncertainites ([FORMULA] standard intervals) as derived by the [FORMULA] method. Column 10: intrinsic dispersion of the FP, [FORMULA], projected along the [FORMULA] direction. The values were obtained by the [FORMULA] fit.

The FP parameters, [FORMULA], [FORMULA] and [FORMULA] were drawn from literature. The photometry is in Gunn [FORMULA] for JFK data and in Kron-Cousins R band for the other samples. For the present study, the differences of the two wavebands can be completely neglected (see Pahre et al. 1998 and Scodeggio et al. 1998). Only galaxies with [FORMULA] were considered (see e.g. Dressler et al. 1987).

The FP coefficients were determined by the [FORMULA] [FORMULA] [FORMULA] and [FORMULA] fits (see Sect. 2.1), by taking into account the measurement errors on the variables as described in Sect. 2.2.

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 [FORMULA] CL for a normal bi-variate. This is accounted for in Fig. 10.

[FIGURE] Fig. 10. FP slopes, a and b, of the samples listed in Table 2. The ellipses define a [FORMULA] CL. The coefficients were derived by the [FORMULA] fit.

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 [FORMULA] and [FORMULA] fits are also listed in Table 2.

[FIGURE] Fig. 11. FP slopes of the clusters listed in Table 2. The coefficients a and b of the different MIST fits are plotted on horizontal panels. The fitting methods are indicated shortly in the lower right-hand corners of the panels on the left. The sample size increases from the left to the right of each panel as shown on the x-axes of the lower panels. The error bars define a [FORMULA] CL. The values of a and b after the magnitude-completeness corrections are also plotted by asterisks. The dashed lines correspond to the weighted means of the slopes.

All the samples (except one) have small size, [FORMULA], so that, as discussed in Sect. 3.2, an accurate estimate of the uncertainties on a and b is not achievable. Approximate error bars were derived by theoretic formulae (Eqs. B17), with the prescriptions of Sect. 3.2.1 to obtain a [FORMULA] CL. Since half of the clusters have [FORMULA], effective theoretic intervals should in fact be more suitable than re-sampling estimates (see Sect. 3.2.2).

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 [FORMULA] of the a and b values. Moreover, they could not represent reliable estimates.

It is also worth to notice that in the [FORMULA], [FORMULA] and [FORMULA] fits, the coefficient a of the largest sample (the Coma cluster) is systematically higher with respect to the other determinations. A weak correlation between a and the sample size seems also to be present.

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 [FORMULA] as the dependent variable, that should be less sensitive to the photometric completeness (see Hudson et al. 1997, hereafter HLS97). By looking at Fig. 11, we see that in the [FORMULA] method there is no systematic difference between Coma and the other samples.

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 [FORMULA] variable for [FORMULA] fit. For each sample we derived the rms of the [FORMULA] residuals. The amount of scatter due to the measurement errors was then subtracted in quadrature. The measurement error scatter projected on [FORMULA] was found to be [FORMULA]. By constructing FP simulations with [FORMULA], the `intrinsic rms values' were corrected for the bias due to the [FORMULA] cut on the cluster samples. The correction was found to be very small ([FORMULA] - [FORMULA]) and largely independent of the simulation parameters.

The [FORMULA] dispersions are shown in Table 2. The mean values of the [FORMULA] intrinsic scatter amounts to [FORMULA], that corresponds to [FORMULA] in [FORMULA].

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 [FORMULA] corrections. The [FORMULA] bias on the slopes of the [FORMULA] fit was estimated from the simulations (see above). It amounts to [FORMULA] for a and is completely negligible for b.


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[FORMULA] intervals).

The difference between the various MIST fits amounts to [FORMULA] for the coefficient a and to [FORMULA] for b. The coefficient a varies from 1.25 to 1.45, and b from 0.33 to 0.35.

The bias that could be introduced by neglecting the measurement errors was found to be negligible for b ([FORMULA]). Concerning the coefficient a, the bias depends on the fitting procedure: it amounts to [FORMULA] for the [FORMULA] and [FORMULA] methods (in agreement with the estimate given by JFK96), and to [FORMULA] and [FORMULA] - [FORMULA] for the [FORMULA] and [FORMULA] fits. Because of the correlation between the uncertainties on [FORMULA] and [FORMULA], it turns out that the [FORMULA] method is quite insensitive to the uncertainties on the FP parameters.

For what concerns the covariance term of the photometric parameters, as discussed in Sect. 2.2, it was estimated by the relation [FORMULA] with [FORMULA] (see JFK95a). By analyzing the correlation of the errors on [FORMULA] and [FORMULA] for various photometric data set drawn from literature, we obtained values of [FORMULA] included between 0.26 and 0.34. Varying [FORMULA] in this range for the calculation of [FORMULA], very small variations ([FORMULA]) were obtained in the FP coefficients. The higher systematic difference, [FORMULA] - [FORMULA], was found for the coefficient a in the [FORMULA] fit.

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 [FORMULA] uncertaintities added in quadrature. The values obtained by JFK96 appear to be consistent with our estimates for each cluster. However, some systematic difference exists. The values of a obtained by the [FORMULA] and [FORMULA] fits are systematically lower than those of JFK96. A small systematic difference is also found for the [FORMULA] method. For the [FORMULA] fit the MIST estimates are systematically higher.

[FIGURE] Fig. 12. Comparison of the FP slopes obtained by the MIST fits with the values of JFK96 (see text). Symbols for the different MIST fits are as in Fig. 1.

The values of b agree with those by JFK96 for the [FORMULA], [FORMULA] and the [FORMULA] methods. On the other hand, the values of the [FORMULA] fit are systematically higher.

The values of the FP slopes obtained by JFK96 for the whole cluster sample, [FORMULA] and [FORMULA], are generally consistent with the mean values reported in Table 3. However, systematic differences are particularly evident for the value of a in the [FORMULA] fit and for the value of b in the [FORMULA] method.

HLS97 derive the FP slopes for a sample of seven clusters of galaxies. By adopting the [FORMULA] method on all the clusters simultaneously, they obtain [FORMULA] and [FORMULA]. These values are consistent with the mean values of the [FORMULA] fit (see Table 3).

The slopes of the individual clusters of HLS97 are derived by a bi-dimensional fit, adopting as independent variable the combination of [FORMULA] and [FORMULA] obtained from the global fit. A direct comparison of our estimates with the individual results of HLS97 is thus not possible.

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.

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Online publication: October 30, 2000