Fundamental plane (elliptical galaxies)
Encyclopedia
The fundamental plane is 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.
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.
Effective radius
The effective radius of a galaxy is the radius at which one half of the total light of the system is emitted interior to this radius. This assumes the galaxy is circularly symmetric...
, average surface brightness
Surface brightness
The overall brightness of an extended astronomical object such as a galaxy, star cluster, or nebula, can be measured by its total magnitude, integrated magnitude or integrated visual magnitude; a related concept is surface brightness, which specifies the brightness of a standard-sized piece of an...
and central velocity dispersion
Velocity dispersion
In astronomy, the velocity dispersion σ, is the range of velocities about the mean velocity for a group of objects, such as a cluster of stars about a galaxy...
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.
Motivation
Many characteristics of a galaxy are correlated. For example, as one would expect, a galaxy with a higher luminosityLuminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...
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:- Larger galaxies have fainter effective surface brightnesses. Mathematically speaking:
(Djorgovski & Davis 1987) where 
is the effective radius, and 
is the mean surface brightness interior to 
. - As
, we can substitute the previous correlation and see that 
and therefore: 
meaning that more luminous ellipticals have lower surface brightnesses.
- More luminous elliptical galaxies have larger central velocity dispersions. This is called the Faber-Jackson relationFaber-Jackson relationThe Faber–Jackson relation is an early empirical power-law relation between the luminosity L and the central stellar velocity dispersion \sigma of elliptical galaxies, first noted by the astronomers Sandra M. Faber and Robert Earl Jackson in 1976...
(Faber & Jackson 1976). Analytically this is:
. This is analogous to the Tully-Fisher relation for spirals. - If central velocity dispersion is correlated to luminosity, and luminosity is correlated with effective radius, then it follows that the central velocity dispersion is positively correlated to the effective radius.
Usefulness
The usefulness of this three dimensional space
is most practical when plotted as 
against 
. The equation of the regression line through this plot is:
Thus by measuring observable quantities such as surface brightness and velocity dispersion (both independent of the observers distance to the source) can provide the effective radius (measured in kpcParsecThe parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....
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
is the diameter within which the mean surface brightness is 
. This relationship has a scatter of 15% between galaxies. In 1991 Gudehus found that the 
correlation exhibits a systematic bias both in distance and absolute magnitude.
- More luminous elliptical galaxies have larger central velocity dispersions. This is called the Faber-Jackson relation