Fundamental plane (elliptical galaxies)

The fundamental plane is a set of bivariate correlations connecting some of the properties of normal elliptical galaxies. Some correlations have been empirically shown.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.The following correlations have been empirically shown for elliptical galaxies:The usefulness of this three dimensional space ( log ⁡ R e , ⟨ I ⟩ e , log ⁡ σ o ) {displaystyle left(log R_{e},langle I angle _{e},log sigma _{o} ight)}   is studied by plotting log R e {displaystyle log ,R_{e}}   against log ⁡ σ o + 0.26 μ B {displaystyle log sigma _{o}+0.26,mu _{B}}  , where μ B {displaystyle mu _{B}}   is the mean surface brightness ⟨ I ⟩ e {displaystyle langle I angle _{e}}   expressed in magnitudes. The equation of the regression line through this plot is:An early use of the fundamental plane is the D n − σ o {displaystyle D_{n}-sigma _{o}}   correlation, given by:The observed empirical correlations reveal information on the formation of elliptical galaxies. In particular, consider the following assumptionsDiffuse dwarf ellipticals do not lie on the fundamental plane as shown by Kormendy (1987). Gudehus (1991) found that galaxies brighter than M V = − 23.04 {displaystyle M_{V}=-23.04}   lie on one plane, and those fainter than this value, M ′ {displaystyle M'}  , lie on another plane. The two planes are inclined by about 11 degrees.