The Faber-Jackson relation in ellipticals

The Faber-Jackson law, Figure 10.3, relates the velocity dispersion $\sigma$ of the stars in an E-type galaxy with the total luminosity $L$,

$$L\propto \sigma^4 .$$ (10.7)

Compare with the Tully-Fisher relation for spirals, eq. (10.1): the velocity dispersion $\sigma$ takes the role of the circular velocity $V_c$.

Comparison of Figure 10.3 with Figure 10.2 shows that the scatter in the Faber-Jackson relation is quite a bit bigger than the scatter in the Tully-Fisher relation: at a given value of $\sigma$, there is a range of $\pm 2$ magnitudes in $M_{\rm B}$ (i.e. a factor 2.5 in $L$) for a given value of $\sigma$, versus a few tenths of a magnitude for a given value of $V_c$ in the Tully-Fisher relation. So ellipticals follow a similar relation as spirals, with more luminous and hence more massive, ellipticals having a large velocity dispersion, but the relation is not very tight.

Figure 10.3: B-band absolute magnitude $M$ for a sample of elliptical galaxies plotted versus the velocity dispersion $\sigma$ (in km s$^{-1}$). Brighter ellipticals galaxies (more negative $M_{\rm B}$; recall that magnitude $M$ is related to luminosity $L$ as $M=-2.5\log(L)$+constant) have a larger velocity dispersion.

From all the parameters we can measure for an elliptical, the total luminosity $L$, the velocity dispersion $\sigma$, the effective radius $R_e$ and the surface brightness $I_e$ (which both enter in the de Vaucouleurs profile, Eq. (3.2)), only three are independent. So we can try to obtain a tighter relation by introducing a second parameter in Eq. (10.7), for example $R_e$. If the galaxy is in virial equilibrium, then its kinetic energy $M \sigma^2$ will be proportional to its potential energy $M^2/R_e$, and so we expect

$$\sigma^2\propto {M\over R_e} .$$ (10.8)

Introducing again the mass-to-light ratio, [M/L] this can be written as

$$\sigma^2 \propto \hbox{[M/L]}{L\over R_e} \propto {L^{1+\alpha}\over R_e} ,$$ (10.9)

where I've assume [M/L] to depend on $L$ as

$$\hbox{[M/L]}\propto L^\alpha .$$ (10.10)

The surface brightness $I_e\propto L/R_e^2$, hence

$$R_e\propto \sigma^{2/(1+2\alpha)} I^{-(1+\alpha)/(1+2\alpha)} .$$ (10.11)

Figure 10.4: Scale radius $R_e$ versus $\sigma^{1.24} I_e^{-0.82}$ for the same sample of Es plotted in Figure 10.3. The galaxies follow this relation with much less scatter then the Faber-Jackson relation in Fig.10.3.

Figure 10.4 shows that

$$R_e\propto \sigma^{1.24} I_e^{-0.82} ,$$ (10.12)

for the same sample of ellipticals plotted in Figure 10.3. The galaxies follow this relation with very little scatter. Comparing the last two equations shows that if we assume $\alpha\approx 0.25$, then $R_e\propto \sigma^{1.33} I_e^{-0.82}$ - very nearly the same to what the figure suggests. Put differently, if the mass-to-light ratio of ellipticals depends on luminosity as $M/L\propto L^{0.25}$, then we can understand why ellipticals follow the relation plotted in Figure 10.4 so tightly.

The relation Eq. (10.12) is called the fundamental plane of elliptical galaxies. In four dimensional $L$, $\sigma$, $R_e$ and $I_e$ space, galaxies do not occupy the whole space, but are restricted to a 3-dimensional surface defined by relation (10.12).