The fundamental plane is a set of bivariate correlations connecting some of the properties of normal elliptical galaxies. Some correlations have been empirically shown. The fundamental plane is usually expressed as a relationship between the effective radius, average surface brightness and central velocity dispersion of normal elliptical galaxies. Any one of the three parameters may be estimated from the other two, as together they describe a plane that falls within their more general three-dimensional space. Properties correlated also include: color, density (of luminosity, mass, or phase space), luminosity, mass, metallicity, and, to a lesser degree, the shape of their radial surface brightness profiles.

Motivation

Many characteristics of a galaxy are correlated. For example, as one would expect, a galaxy with a higher luminosity has a larger effective radius. The usefulness of these correlations is when a characteristic that can be determined without prior knowledge of the galaxy's distance (such as central velocity dispersion – the Doppler width of spectral lines in the central parts of the galaxy) can be correlated with a property, such as luminosity, that can be determined only for galaxies of a known distance. With this correlation, one can determine the distance to galaxies, a difficult task in astronomy.

Correlations

The following correlations have been empirically shown for elliptical galaxies:

Usefulness

The usefulness of this three dimensional space is studied by plotting \log , R_e against, where \mu_B is the mean surface brightness expressed in magnitudes. The equation of the regression line through this plot is: or Thus by measuring observable quantities such as surface brightness and velocity dispersion (both independent of the observer's distance to the source) one can estimate the effective radius (measured in kpc) of the galaxy. As one now knows the linear size of the effective radius and can measure the angular size, it is easy to determine the distance of the galaxy from the observer through the small-angle approximation.

Variations

An early use of the fundamental plane is the correlation, given by: determined by Dressler et al. (1987). Here D_n is the diameter within which the mean surface brightness is 20.75 \mu_B. This relationship has a scatter of 15% between galaxies, as it represents a slightly oblique projection of the Fundamental Plane. Fundamental Plane correlations provide insights into the formative and evolutionary processes of elliptical galaxies. Whereas the tilt of the Fundamental Plane relative to the naive expectations from the Virial Theorem is reasonably well understood, the outstanding puzzle is its small thickness.

Interpretation

The observed empirical correlations reveal information on the formation of elliptical galaxies. In particular, consider the following assumptions These relations imply that, therefore and so. However, there are observed deviations from homology, i.e. with \alpha=0.2 in the optical band. This implies that so so that. This is consistent with the observed relation. Two limiting cases for the assembly of galaxies are as follows. The observed relation lies between these limits.