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If the TF relation has been the workhorse of modern velocity field studies, the Dn-sigma relation has been a short step behind. The closest analogue to the TF relation for elliptical galaxies is actually the predecessor of Dn-sigma, the Faber-Jackson (FJ) relation. FJ expresses the power-law correlation between an elliptical galaxy's luminosity and its internal velocity dispersion,

Equation 6 (6)

where the exponent alpha was found empirically to be ~ 4 ± 1 (Faber & Jackson 1976; Schechter 1980; Tonry & Davis 1981). Although discovered around the same time, and viewed as closely related in physical origin, TF and FJ were not considered equivalently good distance indicators. It was clear from the outset that the scatter in the FJ relation was about twice that of the TF relation, on the order of 0.8 mag. Thus, while the TF relation flourished in the early 1980s as a tool of distance measurement (Section 3), elliptical galaxy surveys focused more on the stuctural and dynamical implications of the FJ relation.

These surveys bore unexpected fruit, however, in the latter part of the 1980s. Two groups conducting surveys of ellipticals arrived independently at a new result: the FJ correlation could be considerably tightened by the addition of a third parameter, namely, surface brightness (Djorgovski & Davis 1987; Dressler et al. 1987). In its modern incarnation, the new correlation has become known as the Dn-sigma relation: a power-law correlation between the luminous diameter Dn-sigma and the internal velocity dispersion sigma,

Equation 7 (7)

where gamma = 1.20 ± 0.10 (Lynden-Bell et al. 1988). (Dn-sigma is defined as the diameter within which the galaxy has a given mean surface brightness. As such, it implicitly incorporates the third parameter into the correlation.)

More broadly, Dn-sigma and its variants may be viewed as manifestations of the Fundamental Plane (FP) of Elliptical Galaxies, a planar region in the three-dimensional space of structural parameters in which normal ellipticals are found. One expression of the FP relates effective diameter to internal velocity dispersion and central surface brightness,

Equation 8 (8)

An early determination of the parameters alpha and beta using B-band photometry gave alpha appeq 1.4, beta appeq 0.9 (Faber et al. 1987). More recently, Bender et al. (1992) found alpha = 1.4, beta = 0.85 using B-band data for a sample of Virgo and Coma cluster ellipticals; the upper panel of Figure 4 shows the FP for this sample. A recent R-band FP analysis by the EFAR group (Wegner et al. 1996) is alpha = 1.23, beta = 0.72. A measurement based on Gunn R-band photometry (Jorgensen et al. 1996) yields a similar value of alpha (1.24 ± 0.07) but a somewhat different value of beta (0.82 ± 0.02), perhaps due to the slightly different bandpass used. Pahre et al. (1995) have recently carried out the first analysis of the FP using K-band photometry, finding alpha = 1.44 ± 0.04, beta = 0.79 ± 0.04.

Figure 4a
Figure 4b
Figure 4. Two versions of the Fundamental Plane for the Virgo and Coma ellipticals studied by Bender et al. (1992). Further details are given in the main text. (The data used for these figures were kindly provided by D. Burstein.)

The two-dimensionality of the loci in parameter space occupied by ellipticals actually makes the FP relations, including Dn-sigma, somewhat less mysterious than the one-dimensional TF sequence. As noted by Faber et al. (1987), such two-dimensionality is expected on virial equlibrium grounds alone. Unlike the TF relation, therefore, the FP is not obviously related to the relative distribution of luminous and dark matter. If the virial theorem were truly all there was to the FP, however, one would find re propto sigmae2 Ie-1. The fact that the FP coefficients differ significantly from these values implies that the mass-to-light (M / L) ratios of ellipticals vary slowly as a function of mass. In particular, the observed FP relations indicate that

Equation 9 (9)

with epsilon appeq 0.15-0.20. Bender et al. (1992) have used this fact to look at the FP in a different way. They define coordinates (kappa1, kappa2, kappa3), each of which is a normalized linear combination of log(sigmae), log (re), and log (Ie). The definitions are such that kappa1 is proportional to log (M) and kappa3 is proportional to log (M/L). A plot of kappa3 versus kappa1, shown in the bottom panel of Figure 4, is very nearly an edge-on projection of the FP. The line drawn through the data is kappa3 = 0.15kappa1 +0.36. A kappa-space analysis of the properties of elliptical galaxies may provide greater insight into the physical processes that shaped them (Bender et al. 1993).

The discovery of the fundamental plane relations was crucial to the use of ellipticals in peculiar velocity surveys because of their much increased accuracy over Faber-Jackson. For example, Jorgensen et al. (1996) estimate that the scatter in log (Re) at fixed sigmae and Ie is 0.084 dex. This corresponds to a distance error of just over 19%, quite comparable to recent estimates of the TF distance error (Section 3). Moreover, Jorgensen et al. have found that the distance error is reduced to 17% when galaxies with sigmae < 100 km s-1 are excluded. This effect is reminiscent of the increased TF scatter at lower velocity widths discussed in Section 3, and may arise for the same reason. Pahre et al. (1995) find a distance error of 16.5% from the K-band FP.

The Dn-sigma relation occupies a special place in the history of peculiar velocity surveys because it was used in the first detection of very large-scale streaming by the ``7-Samurai'' group (Dressler et al. 1987; Lynden-Bell et al. 1988). In the 7-Samurai survey, a full-sky sample of elliptical galaxies revealed a streaming motion of amplitude ~ 500 km s-1 that was coherent across the entire sky to a depth of ~ 40 Mpc. Subsequent studies of spiral galaxies (Willick 1990; Han & Mould 1992; Mathewson et al. 1992; Courteau et al. 1993) have lent confirmation to this result, although the coherence length of the flow remains controversial. Since the late 1980s no new results concerning the peculiar velocity field have been obtained using elliptical galaxy data. However, this situation will change in the coming years as several large surveys of elliptical galaxies (e.g., Wegner et al. 1996) come to fruition.

Like the TF relation, the FP relations are now being studied at appreciable redshift as well. Recently, Bender et al. (1996) have studied a sample of cluster ellipticals at z = 0.37. They have found evidence for mild (~ 0.5 mag) evolution toward brighter magnitudes at such redshifts, comparable to the result found by Vogt et al. (1996) for the TF relation.

Because it is difficult to find Cepheids in nearby elliptical galaxies, there has been little attempt to provide absolute calibrations of the Dn-sigma and FP relations. As a result, elliptical galaxy distances have not figured prominently in the Hubble constant problem. However, this situation may change in the near future, if Surface Brightness Fluctuation distances (discussed in the next Section) can provide an absolute calibration for the Dn-sigma and FP relations.

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