THE FUNDAMENTAL PLANE AND THE SURFACE BRIGHTNESS TEST FOR THE EXPANSION OF THE UNIVERSE

We have deter-mined the Petrosian radius, r(eta), and the enclosed mean surface brightness within the Petrosian radius, [mu]eta, for 33 elliptical and S0 galaxies in the Coma cluster from new accurate CCD surface photometry. For the Petrosian parameter eta = 1.39, r(eta) and [mu]eta are compared with the effective radius, r(e), and the effective mean surface brightness, [mu]e, derived from fitting a de Vaucouleurs law. The fundamental plane (FP) expressed using r(eta) and [mu]eta is the same as the FP found by Jorgensen, Franx, & Kjaergaard (1993) using r(e) and [mu]e. The FP can be used to predict the mean surface brightness within the effective radius or the corresponding Petrosian radius (eta = 1.39) with an uncertainty of +/-0.14 mag for Coma cluster ellipticals. Thus the FP, applied to clusters, appears to be a suitable tool for performing the surface brightness test (SBT) for the expansion of the universe. We suggest that instead of correcting individual galaxies to some standard conditions, e.g., the same metric radius, the fundamental plane itself should be considered the standard. It is argued that the metric size enclosing around 75% of the total light represents a reasonable compromise between resolution and faint level detection when performing the SBT. This radius could be derived as the Petrosian radius corresponding to eta = 2.0 or from a global fit to that part of the observed profile which encompasses 75% of the total light. In case both small and large galaxies are well described by a de Vaucouleurs law the global fit can be performed on a smaller central part of the brightness profile. The use of the FP involves the time consuming determinations of velocity dispersions. We find that [mu]eta (eta = 1.39) can be predicted from the log r(eta) alone with an accuracy of 0.3 mag for the Coma cluster ellipticals. Our discussion of the various error contributions to the predicted mean surface brightness for faint cluster ellipticals at redshifts z < 0.5 shows that the final error is probably dominated by extra scatter due to, e.g., environmental and evolutionary effects. Thus it might be possible that the use of velocity dispersions are not necessary. To get significant results for the SBT, clusters out to a redshift of approximately z = 0.3 have to be observed. For the most distant galaxies light levels down to about 25-26 mag arcsec-2 in the red and sizes as small as approximately 2'' have to be accurately measured. We outline an observational program which will allow the control of the different sources of scatter, including cosmic evolution, producing conclusive results about the expansion of the universe.