3.3.3. Characteristic Parameters of Elliptical Galaxies
The pioneering study of characteristic parameters of elliptical galaxies was made by Fish (1964). For 29 ellipticals with photometry available he found that the total luminosity L re2. Values of Ie were not generally available, but since L Ie re2 for an r1/4 law, we infer that Ie = constant for all galaxies. Under the assumption that the mass-to-light ratio M/L is also constant, Fish derived the dynamically interesting consequence, namely that the potential energy M3/2. He then explored the physical processes that went on during galaxy formation, concluding that the collapse of gaseous protogalaxies was halted by the development of internal opacity.
Better and more extensive data are now available. These show that the above relations need revision. Ie is not the same for all galaxies. Instead, there is a good correlation between Ie and re, shown in Figure 13. This has the form (Kormendy 1980)
The parameter along this line is absolute magnitude MB; lines of constant MB have slope 5 in Figure 13. On average, more luminous galaxies have larger re and fainter Be (see also Oemler 1976; Strom and Strom 1978a, b, c, 1979b). Using MB = Be - 5 log re + constant gives
The scatter about those relations is large, since lines of constant MB are almost parallel to the Be - log re relation. The implications of this result on theories of galaxy formation have not been explored. Some recent work along the lines of Fish's study has been published by Saito (1979), and Silk and Norman (1981).
Figure 13. Be versus log re for 29 elliptical galaxies in poor clusters, from photometry by Kormendy (1977a), King (1978) and Williams and Schwarzschild (1979a, b). Three S0 galaxies with almost negligible disks are included. The straight line is a least-squares fit (equation 6) to these points. Also shown are parameters for cD galaxies and for ellipticals in the cluster A1314 (Oemler 1976). Oemler's V magnitudes have been transformed to B by assuming B-V = 1.00. All r1/4-law parameters discussed in this paper refer to the mean profile, i.e., the profile along a line oriented at 45° to the major axis. Distances to nearby ellipticals have been derived using H0 = 50 km s-1 Mpc-1; relative distances to A1314 and to the cDs are calculated assuming a Hubble constant larger by a factor of 1.46 (Aaronson et al. 1980), corresponding to log re = 0.165. This figure is taken from Kormendy (1980).
The above discussion is unchanged from Kormendy's (1980) review. One recent development is a refined error analysis of the Be - log re relation (Boroson and Kormendy 1982). This is motivated by the fact that Be and re are coupled, a point emphasized by Schechter (1981). The reason for the coupling is again that E-galaxy profiles are nearly power laws. Boroson and Kormendy have derived "2 ellipses", which represent constant root-mean-square deviations of r1/4 laws fitted to the profiles. The 2 ellipses in the Be - log re plane are very elongated and have a slope of ~ 5. That is, the coupling tends to preserve MB. A few of the largest 2 ellipses have lengths which are nearly one-quarter of the parameter range seen in Figure 13. However, most are much smaller than this. It is clear that the parameter coupling has only a small effect on the Be - log re relation.
However, if the fits are allowed to be, less precise, then the 2 ellipses grow very rapidly. Thus large fitting errors tend to scatter the points in the Be - log re diagram along a line of slope ~ 5 (the slope of the 2 ellipses). This may account for the fact that Strom and Strom (1978a, b, c) observe slopes between 3.3 and 5: they fit r1/4 laws between an inner radius limit and an outer brightness limit, even in tidally distended or truncated galaxies (section 3.3.5), without examining the quality of the fits.