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 $L$), 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 as tight as the corresponding $V_c(L)$ relation in spirals.

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 intensity $I_e$ (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 $R_e$. This is how it goes: suppose that 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 assumed [M/L] to depend on $L$ as

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

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

$$R_e\propto \sigma^{2/(1+2\alpha)}\, I_e^{-(1+\alpha)/(1+2\alpha)}\,.$$ (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 much better, 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 which was 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 as 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), hence the name fundamental plane.