Faber-Jackson relationship

The Faber-Jackson relationship (after Sandra M. Faber and Robert Earl Jackson , who discovered the relationship in 1976) is an observed relationship between luminosity  L and velocity dispersion  in elliptical galaxies . According to this, the luminosity depends proportionally on a power of the velocity dispersion: ${\displaystyle \sigma }$

${\displaystyle L\propto \sigma ^{\gamma }}$

The exponent  is very close to 4. ${\displaystyle \gamma }$

The Faber-Jackson relationship is to be understood as the projection of the fundamental plane of elliptical galaxies in order to determine the luminosity and finally the velocity dispersion of an elliptical galaxy even at an unknown distance from elliptical galaxies.

Derivation

One can easily estimate the shape of the Faber-Jackson relationship under certain idealizing assumptions. This gives the exponent of the relationship to . The exponent actually observed depends on the course of the density and the mass-luminosity ratio and deviates more or less strongly from the theoretical value. ${\displaystyle \gamma =4}$

The potential energy of a self-gravitating mass distribution of radius R and mass M is

${\displaystyle U=-{\frac {3}{5}}{\frac {GM^{2}}{R}}}$

The total kinetic energy is

${\displaystyle T={\frac {1}{2}}M\sigma ^{2}}$

With the virial theorem ( ) it follows ${\displaystyle 2T+U=0}$

${\displaystyle \sigma ^{2}={\frac {3}{5}}{\frac {GM}{R}}}$.

If mass and luminosity are proportional to each other , M can be replaced and still has ${\displaystyle M\propto L}$

${\displaystyle L\propto {\frac {\sigma ^{2}R}{G}}}$,

a relationship between R and the velocity dispersion:

${\displaystyle R\propto {\frac {LG}{\sigma ^{2}}}}$.

With a constant surface brightness

${\displaystyle B={\frac {L}{4\pi R^{2}}}}$

follows

${\displaystyle L=4\pi R^{2}B}$,

${\displaystyle L\propto 4\pi \left({\frac {LG}{\sigma ^{2}}}\right)^{2}B}$,

And finally, the relationship we are looking for between luminosity and velocity dispersion:

${\displaystyle L\propto {\frac {\sigma ^{4}}{4\pi G^{2}B}}\propto \sigma ^{4}}$,

swell

• Original work by Faber and Jackson, bibcode : 1976ApJ ... 204..668F

See also

• Tully-Fisher relationship