2.1.2. Rotaional properties
It used to be widely believed that elliptical galaxies were oblate spheroids and that their flattening was due to rotation, in analogy with the rotating figures of equilibrium of incompressible fluids (e.g., Chandrasekhar, 1969). The relationship between rotational velocity and ellipticity of an oblate spheroid with constant ellipticity is (Binney (1978)),
where e2 = 1 - (1 - )2, v is the rotational velocity assumed to be constant and p is the projected velocity dispersion also assumed to be constant. (1) The above relationship has been shown by Binney to apply independently of the form of the density profile.
Figure 2.2. The ratio of the maximum rotational velocity vm to the mean velocity dispersion in the central regions plotted against ellipticity. The open circles show results for elliptical galaxies with absolute magnitudes MBT < - 19.0 and filled circles show results for faint ellipticals with MBT > - 19.0. Crosses show results for the bulges of disc galaxies. The solid line shows the results of Eq. (2.4) which would apply if elliptical galaxies were oblate spheroids with isotropic velocity dispersions. If ellipticals were prolate spheroids with isotropic velocity dispersions about half of the points should lie above the dashed line and half should lie below (adapted from [Davies et al. (1983)]).
In the last few years, rotation curves and velocity dispersion profiles have been obtained for a large number of elliptical galaxies (see e.g. Illingworth, 1981 and references therein), and it is quite clear that most giant ellipticals rotate much more slowly than predicted by Eq. (2.4). The observational results are summarized in Figure 2.2. There are two possible explanations of the results. Ellipticals may be prolate and rapidly rotating, or their flattening may be due to velocity anisotropies rather than rotation. For rapidly rotating prolate spheroids, projection effects will lead to a large spread in v / at any given ellipticity. Binney (1978) has calculated a median curve (shown as the dotted line in Figure 2.2). If ellipticals are prolate and rapidly rotating, about half of the observed points should lie above the line and half should lie below. The observations are inconsistent with this hypothesis, therefore, the flattening of many ellipticals must be due to velocity anisotropies. Recently, it has been found that it is the more luminous elliptical galaxies which rotate slowly (Davies et al., 1983). In Figure 2.3 we show the ratio of the observed value of v / to that predicted on the oblate model (Eq. 2.4) as a function of absolute magnitude. There is a significant correlation, showing that the rotational properties of ellipticals depend on luminosity. A rough approximation to these results is the power-law relation,
where we have taken L* = 1.1 × 1010h-2 L in the B-band (2).
Figure 2.3. The ratio of the observed value of vm / to that predicted on the isotropic oblate model (Eq. 2.4) plotted against absolute magnitude. Results for ellipticals are shown as filled circles whilst crosses show results for the bulges of disc galaxies (adapted from Davies et al., 1983).
1 This equation may be approximated by the simple formula v / p = [ / (1 - )]1/2. Back.
2 Here and throughout this article h denotes Hubble's constant H0 in units of 100 km s-1 Mpc-1. Planck's constant is denoted by hp. Back.