Faber–Jackson relation

The Faber–Jackson relation is a vital mathematical rule in extragalactic astronomy. It describes a specific connection found within elliptical galaxies. These galaxies are large, rounded collections of stars. The relation links a galaxy's luminosity, which is its total light output, to its central stellar velocity dispersion. Velocity dispersion is a measure of how much the speeds of individual stars vary within the galaxy's center. By understanding this link, astronomers can turn a galaxy's internal motions into a way to measure the vastness of space.

To understand how this works, we must look at the physics of gravity and motion. The relation is built upon the virial theorem. This theorem relates the kinetic energy of a system to its gravitational potential energy. In a galaxy, the mass creates a gravitational pull that keeps stars in motion. If a galaxy is very massive, the stars must move at higher speeds to avoid collapsing inward. This higher speed increases the velocity dispersion. Because more mass usually means more stars, a higher velocity dispersion is tied to a higher total luminosity.

Astronomers use this relation as a cosmic yardstick to estimate distances. Measuring how far away a distant galaxy is can be very difficult. However, scientists can use spectroscopy to observe the Doppler shift of light from a galaxy's stars. This shift reveals the central stellar velocity dispersion quite easily. Once this speed is known, the Faber–Jackson relation predicts the galaxy's true luminosity. Astronomers then compare this true luminosity to the apparent magnitude, or how bright the galaxy looks from Earth. This comparison allows them to calculate the distance to the galaxy.

The history of this discovery involves several important scientists. In 1962, Rudolph Minkowski made an early attempt to find a correlation between luminosity and line width. He noted that a relationship existed, but he described it as poor. He suggested that more observations were needed, especially for galaxies with medium brightness. Later, in 1976, astronomers Sandra M. Faber and Robert Earl Jackson presented their formal empirical power-law relation. Their work provided a much stronger mathematical link. They showed that the index for this relation is approximately 4.

Not all elliptical galaxies follow the exact same rule. The relation actually changes depending on the size and brightness of the galaxy. For low-luminosity elliptical galaxies, a team led by Roger Davies found a value of 2 in 1983. In contrast, Paul L. Schechter reported a value of 5 for luminous elliptical galaxies in 1980. This difference exists because massive galaxies grow through homologous merging. Smaller, fainter galaxies form through a different process called dissipation. Because of these different growth paths, the surface brightness of galaxies is not constant.

Some scientists have debated the assumptions used to build these models. In 1972, Allan R. Sandage argued that surface brightness might be constant. However, Donald Gudehus showed in 1975 that these arguments were incorrect. He found that even first-ranked cluster galaxies showed variation in their brightness. This led to a more complex view where the fundamental plane splits into two different planes. These planes are inclined by about 11 degrees from each other. Modern studies also look at how these relations align with theories like Modified Newtonian Dynamics, or MOND.

The Faber–Jackson relation is also a way to study the invisible parts of our universe. It is considered a projection of the fundamental plane of elliptical galaxies. This plane connects several properties, including radius, surface brightness, and velocity dispersion. Scientists like Harry Desmond and Risa H. Wechsler have used dark matter models to test these connections. They want to see if dark matter halos can fully explain why galaxies behave this way. While current models explain some trends, they predict more variation than what we actually see. This suggests there may be hidden relationships between galaxies and dark matter that we do not yet understand.