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3. Kinematics and Dynamics

3.1. The fundamental plane

Giant elliptical galaxies occupy only a small portion of the three-dimensional parameter space defined by the central velocity dispersion (sigma0), effective surface brightness (Ie) and effective radius (re). The manifold occupied by ellipticals is most simply represented as a scaling law

Equation (4)

with A approx 1.4 and B approx -0.9 (Kormendy and Djorgovsky 1989; Bender et al. 1992). Simple arguments from the virial theorem and the above scaling relation suggest that M / L propto M1/6 propto L1/5 (Dressler et al. 1987). Properties of the stellar populations appear to be closely related to structure and dynamics; tight relations also exist between re, Ie and color or metallicity (de Carvalho and Djorgovsky 1989). While the existence of a ``fundamental plane'' for elliptical galaxies provides important clues about the formation of these galaxies, the implications of these clues are currently understood only at a qualitative level (Kormendy and Djorgovsky 1989; Bender et al. 1992).

Several attempts have been made to assess whether dE galaxies lie on the fundamental plane defined by the giants. The task is complicated by the great difficulty in measuring velocity dispersions for faint, low-surface-brightness galaxies. The faint end of the luminosity function is represented by Local Group companions, while the brighter dE's for which velocity dispersions have been measured are mostly members of the Virgo cluster. These latter samples are far from complete, and are biased toward high-surface-brightness objects (Bender and Nieto 1990; Peterson and Caldwell 1993). Furthermore, the prominent nuclei in many of the brighter dE's may be distinct dynamical entities, so using central velocity dispersions may not be sensible if correlations of global parameters are sought.

Nieto et al. (1990) presented the first attempt to extend the fundamental plane to low galactic mass. Their analysis suggested that dE's and also globular clusters fall near or within the fundamental plane, albeit with more scatter than expected from the measurement errors. More recent analysis suggests that dE's do not follow the giant-elliptical scaling relations (Bender et al. 1992; de Carvalho and Djorgovsky 1992; Peterson and Caldwell 1993). Fig. 4 shows the distribution of low-luminosity ellipticals (not all of the diffuse kind) along one projection of the fundamental plane. The local dE's evidently depart from the giant E fundamental plane. The two sequences appear to intersect at MB approx -17, in the regime of the brighter cluster dE's. However, the situation is rendered ambiguous by the large scatter in the dE measurements, and by an apparent inconsistency between the two largest datasets. Bender and Nieto (1990) measured sigma for seven low-luminosity ellipticals. However their sample was manifestly biased toward high-surface-brightness galaxies, and interacting ones at that. Only two of their galaxies fall on the canonical dE sequence, and these had central velocity dispersions of 65 ± 4 and 52 ± 6 km s-1. Peterson and Caldwell (1993) measured eight additional dE's, all but one of the nucleated variety. They found 16.4 < sigma < 39 km s-1. It is not clear whether the discrepancy with Bender and Nieto (1990) is due to the different types of galaxies in the samples, the different spectral resolutions, or different apertures. It is clear that measurements are needed for a complete sample that is unbiased in surface-brightness to determine the true relations for dE's between structure and velocity dispersion. Measurements for cluster dE's with absolute magnitudes fainter than -15, while exceedingly difficult to obtain, are important for testing whether cluster dE's and local dE's really are the same type of object.

Figure 4. One projection of the fundamental plane of elliptical galaxies. Data from Dressler et al. (1987) for Virgo and Fornax cluster ellipticals are shown as x's. Galaxies surveyed by Bender and Nieto (1990) are shown as circles, while those of Peterson and Caldwell (1993) are squares. Local group galaxies are shown by their abbreviations. Data for the Local Group galaxies come from the compilation by Bender et al. (1992), with the exception of Sextans, for which we have used the values in Peterson and Caldwell (1993). The solid line is the least square fit to the Dressler et al. Virgo-cluster data. The dotted line shows the trend for M / L propto L-0.37 predicted by Dekel and Silk (1986). At fixed luminosity, the velocity dispersions reported by Peterson and Caldwell are uniformly lower than those reported by Bender and Neito, rendering it difficult to tell which sequence the cluster dE's actually inhabit.

Dwarf ellipticals appear to define more nearly a one-parameter family, when L, re, Ie, and sigma0 are considered; the scatter in the surface-brightness-magnitude relation (Sect. 2.2.2) is not significantly reduced if surface-brightness is replaced by a linear combination of surface-brightness and log sigma0. However, the paucity of velocity dispersions for galaxies between MB = -16 and MB = -11 renders this to a certain extent an exercise in small-number statistics. The statistics can be improved by substituting color for velocity dispersion. For Virgo and Fornax cluster dE's, there is a small improvement if color is added as an additional parameter, but large scatter remains. The rather heterogeneous samples of galaxies for which colors have been measured seriously compromises the analysis of scaling relations and tests for the number of independent parameters. While giant E samples are reasonably free from selection effects based on color and surface-brightness, dE samples are greatly affected by such selection effects. Low surface-brightness galaxies are typically detected on blue-sensitive emulsions; hence the discovery technique will tend to find blue galaxies and miss red ones if there is a spread in color at fixed bolometric surface-brightness. Furthermore, photometry samples in the literature are a mix of bright, relatively high-surface brightness dE's chosen for ease in getting photometry (Bothun and Caldwell 1984; Caldwell and Bothun 1987), and extremely low surface-brightness dE's chosen because they are interesting (Impey et al. 1988; Bothun et al. 1991).

The analysis of the existing samples suggests correlations that are in reasonable agreement with the predictions of Dekel and Silk (1986). Specifically, while the fundamental-plane relation for giant ellipticals implies M / L propto L1/5, the relation for dwarfs is M / L propto L-0.4, close to the M / L propto L-0.37 relation expected for mass-loss within a dominant dark halo (Dekel and Silk 1986). However, it must be noted that the observed relation has large scatter (and involves many assumptions) and there is a hint of a correlation of M / L for local dE's with position in the outer-galaxy halo (see below).

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