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Plain fundamentals of Fundamental Planes: Analytics and algorithms

The coefficients a and b of the Fundamental Plane relation R ~ Sigma^a I^b 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 logR, logSigma and logI. Our analysis is more generally applicable to any other correlations between three variables: e.g., the color-magnitude-Sigma relation, the L-Sigma-Mbh relation, or the relation between the X-ray luminosity, Sunyaev-Zeldovich decrement and optical richness of a cluster, so we provide IDL code which implements these ideas, and we show how our analysis generalizes further to correlations between more than three variables. We show how to account for correlated errors and selection effects, and quantify the difference between the direct, inverse and orthogonal fit coefficients. We show that the three vectors associated with the Fundamental Plane can all be written as simple combinations of a and b because the distribution of I is much broader than that of Sigma, and Sigma and I are only weakly correlated. Why this should be so for galaxies is a fundamental open question about the physics of early-type galaxy formation. If luminosity evolution is differential, and Rs and Sigmas do not evolve, then this is just an accident: Sigma and I 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-L galaxies evolve more rapidly than do those of higher-L galaxies is able to produce the observed decrease in a (by a factor of 2 at z~1) while having b decrease by only about 20 percent. In such a model, the Mdyn/L ratio is a steeper function of Mdyn at higher z.