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.
