Plain fundamentals of Fundamental Planes: Analytics and algorithms
Ravi K. Sheth (ICTP, UPenn), Mariangela Bernardi (UPenn)

TL;DR
This paper derives explicit formulas for the Fundamental Plane coefficients based on covariances, discusses their generalizations, and explores implications for galaxy formation and evolution.
Contribution
It provides explicit expressions for Fundamental Plane coefficients, generalizes the analysis to other correlations, and discusses implications for galaxy evolution.
Findings
Explicit formulas for a and b in terms of covariances
Generalization to other three-variable correlations
Insights into galaxy luminosity and structural evolution
Abstract
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…
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