Plain fundamentals of Fundamental Planes: analytics and algorithms

Estimates of the coefficients a and b of the Fundamental Plane relation R?sa Ib depend on whether one minimizes the scatter in the R direction, or orthogonal to the plane. We provide explicit expressions for a and b (and confidence limits) in terms of the covariances between log R, log s and log I. Our expressions quantify the origin of the difference between the direct, inverse and orthogonal fit coefficients. They also show how to account for correlated errors, how to quantify the difference between the plane in a magnitude-limited survey and one which is volume limited, how to determine whether a scaling relation will be biased when using an apparent magnitude-limited survey, how to remove this bias and why some forms of the z similar to 0 plane appear to be less affected by selection effects, but that this does not imply that they will remain unaffected at high redshift. Finally, they show why, to a good approximation, the three vectors associated with the plane, one orthogonal to and the other two in it, can all be written as simple combinations of a and b. Essentially, this is a consequence of the fact that the distribution of surface brightness is much broader than that of velocity dispersions, and velocity dispersion and surface brightness are only weakly correlated. Why this should be so for galaxies is a fundamental open question about the physics of early-type galaxy formation. We argue that if luminosity evolution is differential, and sizes and velocity dispersions do not evolve, then this is just an accident: velocity dispersion and surface brightness must have been correlated in the past. On the other hand, if the (lack of) correlation is similar to that at the present time, then differential luminosity evolution must have been accompanied by structural evolution. A model in which the luminosities of low-luminosity galaxies evolve more rapidly than do those of higher luminosity galaxies is able to produce the observed decrease in a (by a factor of 2 at z similar to 1) while having b decrease by only about 20 per cent. In such a model, the dynamical mass-to-light ratio is a steeper function of mass at higher z. Our analysis is more generally applicable to any other correlations between three variables: e.g. the colourmagnitudes relation, the luminosity and velocity dispersion of a galaxy and the mass of its black hole or the relation between the X-ray luminosity, SunyaevZeldovich decrement and optical richness of a cluster. Furthermore, for completeness, we show how our analysis generalizes further to correlations between more than three variables.