Faber-Jackson relation

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The Faber-Jackson relation is an observed relation between the luminosity L and the central velocity dispersion σ in elliptical galaxies. The relation is mathematically expressed as follows:

L \propto \sigma^ \gamma,
where the index γ is known to be very close to 4.


This was first noted by the astronomers Robert Earl Jackson and Sandra M. Faber, in 1976, and is now widely used to obtain important information about the elliptical galaxies.

[edit] A derivation

The gravitational potential of a mass distribution of radius R and mass M is given by the expression:


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


The kinetic energy is
K = \frac{1}{2}M \sigma^2

From the Virial Theorem (2K + U = 0 ) it follows

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

If we assume that the relation among mass and luminosity is a constant, then
\frac{M}{L}=C
we obtain

M \propto L

Getting rid of M
L \propto \frac{\sigma^2R}{G},

so then we have a relation between R and velocity dispersion:
R \propto\frac{LG}{\sigma^2}.

Let us introduce the surface brightness, and assume this is a constant
B=\frac{L}{4\pi R^2},
so then

L = 4πR2B.

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

and finally we obtain the relation between luminosity and velocity dispersion:
L \propto\frac{\sigma^4}{4\pi G^2 B},

that is
L \propto \sigma^4.

[edit] External link

[edit] See also

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