5. Summary and conclusions
The FP of cluster galaxies is one of the most promising tools to study galaxy evolution, to constrain the epoch of galaxy formation, and to set constraints on the geometry of the universe. To those aims the FP has been recently studied in intermediate-redshift () clusters (van Dokkum & Franx 1996, Kelson et al. 1997, Bender et al. 1998, van Dokkum et al. 1998, Jorgensen et al. 1999, Kelson et al. 2000). The main step in the above applications is the comparison between FP determinations in different clusters, wavebands and redshifts, for which the determination of the FP coefficients and of the relative uncertainties plays a crucial rôle.
The present study aims at clarifying various aspects in the determination of the FP coefficients and uncertainties.
5.1. The problem of fitting the FP
We introduced a statistical model of the FP that takes into account the measurement errors on the variables and the intrinsic scatter. The correlation between measurement uncertainties is also accounted for.
We derived the MISTi (Measurement errors and Intrinsic Scatter Three dimensional ) fitting procedures, where is the dependent variable of the fit. A bisector method, , is also introduced.
The assumptions under which the MIST procedures give unbiased estimates of the FP coefficients are that the errors and the intrinsic scatter do not depend on the `location' on the fitting plane (H1), their covariance matrices are known (H2 and H3), and the covariance matrices of the observed quantities are known (H4).
Under these assumptions, the fits give the same coefficients whatever the choice of the dependent variable.
For what concerns the above assumptions, the knowledge of the average measurement errors can be easily introduced in the procedures in order to satisfy H1 and H2, while the problems posed by H4 can be overcome by assuming a normal distribution for the observed quantities.
Assumption H3, on the other hand, cannot be satisfied since we do not have a model for the intrinsic dispersion of the FP variables. We showed that it is precisely the lack of such a model that biases the best-fit coefficients in such a way that their values depend on the choice of the dependent variable and could not coincide with the slopes of the `true relation'.
5.2. Uncertainties on the FP coefficients
We addressed the problem of the estimate of the uncertainties on the FP coefficients by numerical simulations based on the best sampled FP so far (the Gunn r FP of Coma by JFK96).
The results of the simulations are presented in Fig. 4 in terms of and versus . This figure can be used to state the sample size needed to achieve a given accuracy for the FP coefficients.
On the basis of the simulations we tested the performances of the methods used to estimate the uncertainties: the theoretical formulae and the bootstrap technique.
We showed that for the theoretical formulae can recover the variances of the FP coefficients within about 5%, while for the uncertainties are systematic smaller than the true values and, what is more troublesome, are affected by a large scatter.
The bootstrap technique gives, on the average, more accurate estimates than theoretical methods but for small samples the scatter of the individual values is much larger. Again, a minimum sample size of exists for the bootstrap to give reliable estimates.
5.3. Implications for the use of the FP
As an application of the present study we addressed in Sect. 4 the question of the `universality' of the FP. The main obstacles are the limited size of the available (homogeneous) samples and their different completeness characteristics.
We find that the slopes of the FP in the ten clusters considered are consistent within the large uncertainties (due to the small number of galaxies in most of the samples). We stress that this result should be taken at most as an indication in favor of the universality of the FP and not as an evidence.
With the available data samples it is not possible to settle the question of the universality: to this aim, larger and more homogeneous samples would be needed.
For the application of the FP to the study of galaxy evolution it would be crucial to know the behavior of the slopes as a function of redshift. Pahre et al. (1998) built a model for the FP that predicts a change in slope with redshift: if age contributes to the tilt of the FP, massive galaxies should be much older than the least luminous ones. As a consequence of this `differential' evolution, the slopes will change with redshift. They predict the change with redshift of the coefficient a of the FP (see their Fig. 10) by adopting two different models for the stellar evolution (Bruzual & Charlot 1998 and Vazdekis et al. 1996). The Bruzual & Charlot models produces a rapid change of a, that should decrease by about 0.15 from z=0 to z0.3, namely, 12% of its local value in the Gunn band.
Recently, Kelson et al. (2000) derived the FP for a cluster at z=0.33 with 30 galaxies. The authors claim that the slopes are fully consistent with those of the local FP (Coma). They derive the FP by the orthogonal fit and give the following uncertainties for the slopes: =0.1 and =0.11, that are consistent with the values obtained here from the simulations with n=30 (see Fig. 4). These data seem thus to rule out the rapid evolution of a predicted with the Bruzual & Charlot model.
The situation is quite different with the Vazdekis et al. models. At the slope a decreases by only 0.03%. The minimum sample size needed to achieve such a precision at would be , which is out of the reach of any possible observation. The situation improves by moving at higher redshifts, where the magnitude of the predicted change of the slopes is larger. For the cluster at z=0.83 studied by van Dokkum et al. (1998), in fact, the situation is by far more promising. To test the predicted change in slope it would be sufficient to determine the FP on galaxies, that is well within the possibility of a 8-10m class telescope in a reasonable number of nights.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 2000