Kormendy relation

The Kormendy relation is an empirical scaling relation between the effective radius of a galaxy and its effective surface brightness. It is mainly observed among elliptical galaxies and other stellar spheroids. The effective radius, written as $R_{e}$, contains half of a galaxy's total light. At a given photometric wavelength, larger spheroids generally have fainter average surface brightnesses. The relation was presented by astronomer John Kormendy in 1977 while studying the structural properties of galaxies.

Messier 105, an elliptical galaxy. Measurements of an elliptical galaxy's size and surface brightness can be placed on the Kormendy relation.

Form

Surface-brightness profile of the brightest galaxy in Abell 85. Profiles of this kind are used to determine effective radius and effective surface brightness.

A commonly used form of the relation is

$\left\langle \mu _{e}\right\rangle =\alpha +\beta \log _{10}\left({\frac {R_{e}}{\mathrm {kpc} }}\right),$

where $\left\langle \mu _{e}\right\rangle$ is the mean surface brightness within $R_{e}$, usually measured in magnitudes per square arcsecond. The constants $\alpha$ and $\beta$ are the zero point and slope of the fitted relation. Because the magnitude scale is reversed, a larger value of $\left\langle \mu _{e}\right\rangle$ represents a fainter surface brightness. Some studies instead use the surface brightness measured directly at $R_{e}$. The fitted values depend on the wavelength, galaxy sample, redshift, and method used to model the galaxy's light distribution.

Interpretation and use

The Kormendy relation is a two-dimensional projection of the fundamental plane of early-type galaxies. The fundamental plane relates effective radius and surface brightness to the central velocity dispersion of a galaxy's stars, whereas the Kormendy relation uses only photometric measurements. Differences in its slope or zero point can reveal changes in galaxy size, stellar populations, and luminosity. Astronomers therefore compare the relation at different redshifts to study the structural evolution of early-type galaxies.

The relation is also used as a structural test for galactic bulges. Many classical bulges occupy the same region of the relation as elliptical galaxies, while numerous pseudobulges appear as lower-surface-brightness outliers. It is not an exact universal law: dwarf and giant ellipticals may form a curved distribution when a wide range of luminosities is considered, so a linear fit derived for bright galaxies cannot automatically be extended to every spheroidal system.

See also

• Faber–Jackson relation
• Fundamental plane (elliptical galaxies)
• Galaxy scaling relations
• Sérsic profile