In the last 20 years our notions about E galaxies have changed radically; these galaxies are much more interesting -- and complicated -- than they seemed at first.
Traditionally, elliptical galaxies were seen as rather simple systems. Their overall luminosity profiles followed the de Vaucouleurs (1948) law, while their central regions fit the constant-density cores of King (1966) models. Elliptical galaxies were assumed to be oblate spheroids, flattened by rotation. The stars within them were viewed as belonging to a single, ancient population like the bulge and halo populations of our galaxy; gas and dust were thought to be absent. Finally, elliptical galaxies were considered to be dynamically unevolved.
At one level or another, all of the above are incorrect.
In photographic photometry at arc-sec resolution, elliptical galaxies seem to have luminosity profiles which flatten off as the projected radius R -> 0 (King 1978). These luminosity profiles are well-fit by dynamical models based on a `lowered Maxwellian' velocity distribution (King 1966); at small R, such models have projected luminosity profiles resembling
I_0 (1) I(R) = ------------- , 1 + (R/R_c)^2where I_0 is the central surface brightness and R_c is the core radius. However, Schweizer (1979) showed that the galaxies which fit Eq. 1 may be fit just as well by seeing-convolved de Vaucouleurs (1948) laws; as the de Vaucouleurs law rises all the way to R -> 0, the reality of the cores seen in the photometry is open to question.
Ground-based CCD photometry (Lauer 1985, Kormendy 1987) confirmed that the well-resolved cores in some nearby elliptical galaxies are not entirely artifacts of seeing. These galaxies have luminosity profiles which `break' away from a steep slope at a radius comparable to the measured core radius, and the break at least is real. But within these breaks the seeing-deconvolved luminosity profiles continued to slowly rise, contradicting Eq. 1.
HST photometry shows that cores of constant surface density are a myth (Ferrarese et al. 1994, Lauer et al. 1995). In some galaxies the deconvolved luminosity profile rises steeply all the way to the pre-repair resolution limit of about 0.1 arc-sec; these Lauer et al. call `power-law' galaxies since their inner luminosity profiles fit power-laws with logarithmic slopes < -0.5. In others the profile rises steeply to a break, and then more slowly further in; these they call `core' galaxies. This is an bad terminology since none of these galaxies actually have constant-density cores; inside of the break, luminosity profiles rise with logarithmic slopes which are > -0.3 but always negative.
A few E galaxies have nuclear star clusters with densities much higher than the cores they inhabit; some of these nuclei may be rotating, disk-like systems (Kormendy & Djorgovski 1989; hereafter KD89).
Projected axial ratios range from b/a = 1 to 0.3, but not flatter (Schechter 1987). Apparent ellipticity is generally a function of projected radius, with a wide range of profiles (Jedrzejewski 1987).
Isophotal twists are common. It's unlikely that intrinsically twisted galaxies could be dynamically stable, and no galaxy with oblate surfaces of constant stellar density can exhibit a twist in projection. But photometric twists like those observed can result if the surfaces of constant stellar density are triaxial ellipsoids with axial ratios varying with radius (MB81, Fig. 5-27). Thus isophotal twists are generally interpreted as evidence for triaxiality (Kormendy 1982).
As a rule, E galaxies become redder toward their centers; a factor of 10 decrease in radius typically produces an 0.03 change in b-V and a 0.10 change in u-V (KD89).
Absorption features in the integrated spectra of E galaxies also show significant gradients with radius; metal lines become stronger with decreasing radius, while H-beta lines may get stronger or weaker. Line-strength gradients in bright galaxies are smaller than those in faint galaxies but there are large variations from galaxy to galaxy (Gorgas & Efstathiou 1987, Franx et al. 1989).Color and line-strength gradients are both consistent with the interpretation that metalicity is the primary factor varying with radius (Faber 1977). The logarithmic slope of the metal abundance is estimated to be about -0.2 for a typical elliptical (KD89).
Spectral synthesis models indicate that the stars at any given radius span a range in metal abundance (e.g. Pickles 1987). This is consistent with the 2 dex spread in metalicity seen in the bulge of the Milky Way (Whitford & Rich 1983).
Evidence for a spread in ages may be present in the deep Balmer lines seen some systems and in indications of blue light above & beyond what one expects for a purely old population (Pickles 1987).
The rotation velocities of bright E galaxies are much too low to account for the flattenings we observe; fainter E galaxies, however, rotate at about the rates implied by their shapes (Davies 1987).
E galaxies may exhibit minor-axis rotation; more generally, the apparent rotation axis and the apparent minor axis may be misaligned (Franx et al. 1991). While in most galaxies these misalignments are modest, a few galaxies appear to rotate primarily about their minor axes.
The inner regions of elliptical galaxies may exhibit unusual kinematics; for example, about a quarter of all elliptical galaxies have inner regions which counterrotate with respect to the rest of the galaxy (KD89). Although such `kinematically decoupled' regions are generally not photometrically distinct, several such E galaxies have features in their line-strength profiles coincident with the kinematically decoupled regions (Bender & Surma 1992).
Integrated colors of E galaxies show systematic trends with luminosity: Brighter galaxies are redder, by about 0.10 in u-V per magnitude (MB81, Chap. 5-4). As within individual galaxies, it appears that here too the changes in color may be largely explained by changes in metalicity (KD89).
The luminosities of E galaxies are highly correlated with their velocity dispersions; this is the Faber-Jackson relation, generally expressed as a power law,
n (2) L ~ sigma ,where L is the galaxy luminosity, sigma is the central line-of-sight velocity dispersion, and the index n is about 4, but shows significant variations from sample to sample (KD89).
A significant correlation is also seen between the effective (or half-light) radius, R_e, and the surface brightness at that radius, I_e, of the form (KD89)
-0.83 (3) R_e ~ I_e .
Most of the scatter in the Faber-Jackson relationship is intrinsic; it reflects real galaxian properties, not measurement errors. A number of authors attempted to identify the `second parameter' which would account for this scatter.
Modern data show that elliptical galaxies obey a relationship of the form
1.4 -0.9 (4) R_e ~ sigma I_e(KD89). This is called the fundamental plane because in a space whose axes are log R_r, log sigma, and log I_e, elliptical galaxies lie on a rather thin, nearly planar surface. Note that the power-law index of I_e is nearly the same as in Eq. 3.
On dimensional grounds, a self-gravitating system should obey a relationship of the form
2 M (5) sigma ~ G - , Rwhere G is the gravitational constant, M is a characteristic mass, and R is e.g. the half-mass radius. In terms of the system's mass-to-light ratio (M/L), this implies
2 -1 -1 (6) R ~ sigma I_e (M/L) ,a relationship very similar to Eq. 4 (KD89).
The fact that the exponents appearing in Eq. 6 are not quite the same as those in Eq. 4 can be explained if (M/L) is a slowly-varying function of galaxy mass; the observed exponents are obtained from Eq. 6 if (KD89).
0.2 (6) (M/L) ~ M .
The fundamental plane is a valuable tool for estimating distances to galaxies; sigma and I_e are measured from spectroscopy and photometry and used in Eq. 4 to obtain the physical radius R_e in kpc; comparison with the apparent R_e then yields the distance. But if (M/L) is a function of galaxy environment, then this procedure will yield biased estimates of the distance. Some researchers have found evidence that the fundamental plane does depend on cluster richness (KD89).
Due date: 1/23/97
3. Using Eq. 5, the definition of the mass-to-light ratio (M/L), and the assumption that all elliptical galaxies have similar forms, derive Eq. 6.
Last modified: January 17, 1997