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 r_e². Values of I_e were not generally available, but since L I_e r_e² for an r^1/4 law, we infer that I_e = 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 M^3/2. 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. I_e is not the same for all galaxies. Instead, there is a good correlation between I_e and r_e, shown in Figure 13. This has the form (Kormendy 1980)

(6)

The parameter along this line is absolute magnitude M_B; lines of constant M_B have slope 5 in Figure 13. On average, more luminous galaxies have larger r_e and fainter B_e (see also Oemler 1976; Strom and Strom 1978a, b, c, 1979b). Using M_B = B_e - 5 log r_e + constant gives

(7)

The scatter about those relations is large, since lines of constant M_B are almost parallel to the B_e - log r_e 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 B_e - log r_e relation (Boroson and Kormendy 1982). This is motivated by the fact that B_e and r_e 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 r^1/4 laws fitted to the profiles. The ² ellipses in the B_e - log r_e plane are very elongated and have a slope of ~ 5. That is, the coupling tends to preserve M_B. 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 B_e - log r_e 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 B_e - log r_e 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 r^1/4 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.