The fundamental plane (FP) is a bivariate relation between observed global properties of early-type (E) galaxies like the effective radius , the mean surface brightness , and the central velocity dispersion (e.g. Dressler et al. 1987, Djorgovski & Davis 1987 ) of the form 2:
The main features of the FP are its small scatter ( in , of the same order of the measurement errors) and the `tilt' of its slopes with respect to the prediction of the virial theorem for a family of homologous systems with constant ratio.
It is generally believed that the `tilt' carries information on the nature of Es, although no convincing interpretation has been found that can be reconciled with the small scatter (see e.g. Ciotti et al. 1996, Renzini & Ciotti 1993). It is now clear that part of the `tilt' is due to the intrinsic non-homology, both structural and dynamical, of the E family (see e.g. Busarello et al. 1997, Graham & Colless 1996, Prugniel & Simien 1997 and references therein). Stellar population effects (e.g. systematic differences in age/metallicity and interstellar matter content) should account for the remaining tilt (see e.g. Mobasher et al. 1999). In that case, the tilt should decrease when moving to near-infrared wavelengths, where these effects are minimized: some evidences exist that this is actually the case (e.g. Pahre et al. 1998, Scodeggio et al. 1998 and references therein). The wavelength dependence of the FP enters also in the comparisons between FP determinations at different redshifts, where different rest-frame wavebands are sampled.
The FP has been soon recognized as a precise tool for distance determinations (e.g. Dressler et al. 1987, Hudson et al. 1997) and, more recently, for constraining cosmological parameters and for studying galaxy evolution (van Dokkum& Franx 1996, Bender et al. 1998, Jorgensen et al. 1999, Kelson et al. 2000, and references therein).
All the above applications are based on comparisons between different determinations of the FP: e.g. different samples, different wavebands, and different redshifts. The method used to derive the FP coefficients and the relative uncertainties play therefore a central rôle for a proper use of the FP.
The determination of the FP requires considerable observational and data analysis efforts (see e.g. Ziegler & Bender 1997 and Ziegler et al. 1999), it is thus desirable that a similar effort be made for the derivation of the coefficients of the FP and for an accurate estimate of the relative uncertainties.
There is still no agreement in the scientific community about the fitting method to adopt. This is partly due to the numerous applications for which the FP is derived (see above), but it also originates from two important points: (1) the measurement errors on the FP variables are comparable (see e.g. Smith et al. 1997, hereafter SLH97) and correlated (see Jorgensen et al. 1995a, hereafter JFK95a), (2) the scatter around the plane is not completely accounted for by the measurement uncertainties but has also an intrinsic origin (see e.g. Jorgensen et al. 1996, hereafter JFK96).
In the present work we address the problems relative to: (I) the choice of the procedure to derive the FP coefficients (Sect. 2) and (II) techniques to estimate the corresponding uncertainties (Sect. 3).
Starting from the bi-dimensional models introduced by Akritas & Bershady (1996, hereafter AkB96), we propose in Sect. 2.1 statistical models of the FP that account for the various sources of scatter on the variables and derive the relative fitting procedures: the MIST (Measurement errors and Intrinsic Scatter Three dimensional ) fits.
We then discuss in Sect. 2.2 the capability of the fitting procedures to derive the coefficients of the FP.
Sect. 3 deals with the uncertainties of the coefficients of the FP. The simulation algorithm used for the present analysis is described in Sect. 3.1. In Sect. 3.2 the different methods to estimate the uncertainties are analyzed and compared to the results of the simulations.
The results and the techniques developed in Sects. 2, 3, are applied in Sect. 4 to the comparison of the FPs of different clusters of galaxies.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 2000