Annu. Rev. Astron. Astrophys. 1989. 27: 235-277
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8. PARAMETER CORRELATIONS AND SCALING LAWS

One of the main astrophysical uses of surface photometry is for the study of parameter correlations and scaling laws. These contain valuable information about galaxy formation and evolution. Also, correlations between distance-dependent and distance-independent quantities are vital for the mapping of large-scale structure and velocity fields.

8.1. Families of Ellipsoidal Stellar Systems

A fundamental application of parameter correlations has been the demonstration that diffuse dwarf spheroidal (dSph) galaxies are a family of objects unrelated to ellipticals. Baade (1944) long ago noted that NGC 147, NGC 185, NGC 205, and the Galactic dwarf spheroidals form a low- surface-brightness sequence quite unlike ordinary ellipticals. Until the mid-1980s, most people believed that the transition between these sequences is continuous [e.g. Binggeli et al. (1984), but contrast Michard (1979) and Farouki et al. (1983)]. Then, in important and somewhat neglected papers, Saito (1979a, b) pointed out that dSphs have anomalously low binding energies compared to giant ellipticals. He suggested the now-favored explanation that dwarfs have low densities because supernova-driven winds have removed large amounts of gas. Later, Wirth & Gallagher (1984) were the first to emphasize that there are two unrelated sequences of early-type galaxies: the diffuse dwarfs, and an E-galaxy sequence whose low-luminosity end consists of galaxies like M32. They pointed out that the extreme properties of M32-like dwarfs are not due to tidal truncation but are intrinsic to low-luminosity ellipticals (see also Section 7.2). Also, they found additional examples in the Fornax cluster, showing that M32 is not a fluke. The case was further strengthened by Kormendy (1985b, 1987c), who demonstrated a clear separation into two families overlapping in luminosity. The key to this was CFHT seeing good enough to define the low-L end of core parameter scaling laws for ordinary ellipticals. The differences between the families are also seen in global properties (3) (Saito 1979a, Okamura 1985, Dekel & Silk 1986, Ichikawa et al. 1986, 1988, Kormendy 1987c). These, results are not due to selection effects, as suggested by Phillipps et al. (1988). The distribution of parameters for diffuse dwarfs is undoubtedly biased by selection; remarkably low-surface- brightness galaxies are still being discovered (Sandage & Binggeli 1984, Impey et al. 1988). But luminosity "icebergs" hidden under the sky brightness (Disney & Phillipps 1987, 1988) only contribute to the distinction between E and dSph galaxies. Further evidence for this distinction includes a large difference in luminosity functions: Ellipticals have a nearly Gaussian luminosity function that peaks at MB appeq - 18, while dSph galaxies begin to appear at MB appeq - 18 and in Virgo then become more numerous at least as rapidly as L-1.35 [Wirth & Gallagher 1984, Sandage et al 1985a, b, Impey et al. 1988; see Binggeli (1987) for a review]. The distinction between the elliptical and diffuse dwarf galaxy families points to a fundamental difference in formation history.

A clue to the origin of dSph galaxies is provided by the observation that they are structurally similar to dwarf spiral and irregular (dS+I) galaxies (Faber & Lin 1983, Lin & Faber 1983, Caldwell 1983, Wirth & Gallagher 1984, Sandage & Binggeli 1984, Binggeli et al. 1985, Kormendy 1985b, 1987c, Okamura 1985, Binggeli 1985, Ichikawa et al. 1986, 1988, Karachentseva et al. 1987, Impey et al. 1988). They are not merely dS+I galaxies seen between bursts of star formation, because they contain virtually no gas (Bothun et al. 1985, Impey et al. 1988). Two formation mechanisms are discussed at length in the literature. First, the basic low-density structure of dwarf galaxies is probably due at least in part to supernova-driven galactic winds; these can turn some dS+I galaxies into dwarf spheroidals (e.g. Larson 1974, Saito 1979b, Silk 1983, Dekel & Silk 1986, Vader 1986a, 1987, Silk et al. 1987, Yoshii & Arimoto 1987). In addition, there is strong evidence, at least in clusters, that some dwarf spheroidals formed from dS+I galaxies by ram-pressure stripping of their gas [see Lin & Faber (1983), Binggeli (1985), and Kormendy (1987c) for reviews]. There are other possibilities too. Dwarf spheroidals could be dS+I galaxies that turned all of their gas into stars (Kormendy 1985b, Davies & Phillipps 1988, Binggeli et al. 1989). In certain circumstances, dSph galaxies could even turn back into dS+I galaxies by accreting gas (Silk et al. 1987). In the search for simple, unique explanations, we should not forget that all of these things may happen.

8.2. Correlations With Galaxy Luminosity

The results of the previous section were based on correlations of various physical scale parameters with total luminosity L. The best known of these is the Faber-Jackson (1976) relation L propto sigman. The slope is n appeq 4, but with a real variation, depending on the sample definition (Faber & Jackson 1976, Tonry & Davis 1981, Tonry 1981, Terlevich et al. 1981, de Vaucouleurs & Olson 1982, Kormendy & Illingworth 1983, Dressler 1984a). Many of these authors combined their data with estimates of effective radii and found a weak correlation between mass-to-light ratio and luminosity, M/L propto L0.35±0.15. This is also seen in core mass-to-light ratios (K87, Kormendy 1987c).

A correlation between the de Vaucouleurs (1948) effective radius re and surface brightness Ie was found by Kormendy (1977b, 1980, K82); modern data give re propto Ie-0.83±0.08 (e.g. Hoessel & Schneider 1985, Hamabe & Kormendy 1987, Djorgovski & Davis 1987). This implies that more luminous galaxies have larger re and fainter Ie, although with large scatter. Hamabe & Kormendy show that these relations are not significantly affected by the coupling of measurement-errors in the parameters. They are also largely independent of how the parameters are defined; e.g. similar relations hold for core parameters (K82, Kormendy 1984, 1985b, 1987c, K87, Lauer 1985b, 1988b).

An important correlation between luminosity and the dynamical importance of rotation was discovered by Davies et al. (1983). They found that low-luminosity ellipticals and bulges rotate rapidly, have nearly isotropic velocity dispersions, and are flattened by rotation. In contrast, bright ellipticals rotate slowly, are pressure supported, and owe their shapes to velocity anisotropy. Let V/sigma be the ratio of the maximum rotation velocity to a suitable mean velocity dispersion (see Davies et al. 1983). Then the level of rotational support can be parametrized by the ratio (V / sigma)* of V / sigma to the value expected for an isotropic oblate spheroid. The result that (V / sigma)* appeq 1 for faint galaxies and << 1 for bright ones could arise if protoellipticals acquired angular momenta through tidal torques, and if mergers then produced brighter ellipticals in which rotation got scrambled.

8.3. Multiparameter Correlations: the "Fundamental Plane" of Elliptical Galaxies

Multiparameter correlations were discovered through studies of correlated residuals from relations like those of Section 8.2. A breakthrough in our understanding of scaling laws required the appearance of large, homogeneous data sets based mainly on CCD photometry and long-slit spectroscopy. Also, this work has benefited from the application of statistical tools like principal component analysis (PCA) (e.g. Brosche 1973, Bujarrabal et al. 1981, Brosche & Lentes 1983, Lentes 1983, Efstathiou & Fall 1984, Whitmore 1984, Murtagh & Heck 1987). However, in PCA, the astrophysics can get lost in too many eigenvectors. Therefore, simple techniques like bilinear least-squares fits remain useful to provide physical insight.

The presence of intrinsic scatter in the Faber-Jackson relation was correctly interpreted as an indication of a "second parameter." In an important paper, Terlevich et al. (1981) proposed that this second parameter is metallicity, measured by the Mg2 index, and possibly axial ratio. Their results were challenged by Tonry & Davis (1981) and then read-dressed by Efstathiou & Fall (1984). However, relatively poor data sets available at the time did not permit a resolution of the problem. Authors agreed that elliptical galaxies are at least a two-parameter family, but the second parameter could not clearly be identified.

More accurate data confirm that the variance of global properties is exhausted almost entirely by two variables (Tonry & Davis 1981, Lauer 1985b, 1987, Burstein et al. 1986, Djorgovski & Davis 1986, 1987, Dressler et al. 1987, Faber et al. 1987, Djorgovski 1987a, Dressler 1987, de Carvalho & da Costa 1989). These data show that bulges and ellipticals lie in an inclined "fundamental plane" in the space of observed parameters (Figure 2),

R propto sigma1.4±0.15 I-0.9±0.1 .         (1)

Here R can be any consistently defined radius derived from surface brightness profiles, such as the core or effective radius, but not an isophotal radius. An equivalent relation is obtained for luminosity. The old Faber-Jackson and radius-surface brightness relations are projections of the fundamental plane. The luminosity-color and mass-metallicity relations are also contained in the fundamental plane. Its tilt with respect to the planes of observed parameters produces the correlated intrinsic scatter seen in the projected relations.

An alternate form of Equation 1 is the relation between the modified isophotal diameter and velocity dispersion, Dn propto sigma4/3 (Burstein et al. 1986, Dressler et al. 1987). Here Dn is defined as the circular diameter within which the mean surface brightness reaches a certain fiducial value, e.g. µB = 20.75 mag arcsec-2 in the case of Dressler et al. (1987). The above authors show that the Dn - sigma relation is equivalent to Equation 1, provided that all elliptical galaxies have brightness profiles of the same shape.

Figure 2

Figure 2. Projections of the fundamental parameter plane of elliptical galaxies. Top panels: the one-parameter scaling relations discussed in Section 8.2, i.e. (left) the relation between radius and mean surface brightness, and (right) that between luminosity and velocity dispersion (the Faber-Jackson relation). Bottom left: the surface brightness-velocity dispersion correlation is the fundamental plane seen almost face-on. This is an observer's version of the cooling diagram from theories of galaxy formation. Bottom right: this relation between the radius and a combination of surface brightness and velocity dispersion is the fundamental plane seen edge-on. The data are from Djorgovski & Davis (1987). All photometric quantities are in the Lick rG band and are measured at or within the re elliptical isophote. The crosses are median error bars for all points in each panel.

Another alternative representation of the fundamental plane - a relation between radius, surface brightness, and a metallicity indicator (color or Mg2 index) - has been obtained by de Carvalho & Djorgovski (1989). The two-dimensional nature of the manifold of elliptical galaxies implies that there must be a second parameter in the relation between mass and metallicity; this is identified as the luminosity density (S. Djorgovski & R.R. de Carvalho, in preparation).

The residual scatter about the fundamental plane is ~ 20% per galaxy (given as the relative error of distance or radius). It is mostly or entirely due to measurement errors. Any cosmic scatter cannot be larger than a few percent.

A group at the Tokyo Astronomical Observatory has obtained and analyzed photographic surface photometry of galaxies of all Hubble types (Kodaira et al. 1983, Okamura et al. 1984, Watanabe et al. 1985). Based on PCA, they also conclude that there are two dominant dimensions in the parameter space of luminosity, isophotal diameter, surface brightness, and central light concentration. Kodaira (1988) proposes that one of the principal components is phase space density. These results are in agreement with work described above when ellipticals are treated separately. They also agree with results obtained separately for spirals (e.g. Whitmore 1984). Spirals and ellipticals have similar principal-component solutions. Nevertheless, it is not clear whether it is meaningful to lump together galaxies of different Hubble types, since different dynamical subsystems and stellar populations (young disks and old spheroids) contribute to the measured quantities.

In retrospect, the fundamental plane was already implicit in papers by Michard (1979), de Vaucouleurs & Olson (1982), Brosche & Lentes (1983), and Lentes (1983), although they did not recognize the full implications of their results.

There is some controversy about whether luminosity is the "first" parameter (i.e. whether it accounts for the greater part of the variance in other parameters). The present authors disagree on this point. SD believes that it is misleading to consider luminosity as the first parameter. The axis perpendicular to the luminosity in the fundamental plane does not correspond to any direct observable. Even if subsystems of galaxies have a first parameter, this does not prove that the same parameter has the same controlling effect on galaxies as a whole. SD therefore believes that it is most profitable to think of the velocity dispersion and surface brightness as the principal variables from which one can derive luminosity, radius, and other quantities of interest. JK is unconvinced. Even though the <µ>e - sigma diagram in Figure 2 shows no correlation, he worries that subtle problems may have enlarged the scatter. Large samples were required to explore these issues; then many of the galaxies are far away and suffer from problems like seeing and sample selection. Also, cores of nearby galaxies do show a µ0 - sigma correlation (J. Kormendy, in preparation). JK believes that more work is needed on the question of whether one first parameter is more fundamental than the others.

8.4. Uses and Interpretation of the Fundamental Plane

The fundamental plane is a powerful new distance indicator for early-type galaxies. Using it, Lynden-Bell et al. (1988) have discovered large-scale galaxy streaming motions toward the Hydra-Centaurus Supercluster (the "Great Attractor" model).

The plane also contains valuable clues about galaxy formation. Its solutions are very robust, and the residual scatter is very low. The solution has the same form for ellipticals and bulges, it spans about three orders of magnitude in luminosity, it varies little (if at all) in different environments, and it does not depend on how the parameters are measured. It must reflect an important regularity in the process of elliptical galaxy formation or transformation. One useful representation of the fundamental plane is its projection on the log sigma-log I plane of observables. This is the "cooling diagram" in theories of galaxy formation [i.e. virial temperature vs. density (e.g. Rees & Ostriker 1977, Faber 1982, Silk 1983, 1985, 1987, Blumenthal et al. 1984)]. The position of a galaxy in this diagram is related to the amount of dissipation during its formation.

The fundamental plane can be understood using the following simple argument (Faber et al. 1987, Djorgovski et al. 1989). The virial theorem implies that galaxies must satisfy a relation that is very similar to Equation 1:

R ~ kS kE sigma2 I -1 (M / L) -1 .         (2)

The parameter kS reflects the density, luminosity, and kinematic structure of a galaxy; it would be a constant if all galaxies considered had the same dynamical structure. The parameter kE is the ratio of absolute potential energy to kinetic energy for a galaxy: kE > 1 for a bound system, and kE = 2 for a virialized one. The deviations in Equation 1 of the coefficients of sigma and I from 2 and -1, respectively, reflect the dependence of kS kE(M / L) -1 on galaxy mass or other fundamental plane variables. If all of the variation is in mass-to-light ratio, this implies the scaling relation M/L propto M0.2 (Faber et al. 1987, Djorgovski 1987b; cf. Section 8.2). A more complete discussion is given by Djorgovski et al. (1989).

The parameters kS, kE, and M/L depend on the formation and evolutionary histories of galaxies (Djorgovski et al. 1989). Our present understanding of galaxy formation is that it consists of a series of dissipative merging and infall processes, most of which are affected by environment (e.g. Silk & Norman 1981, Silk 1987). In fact, Vader (1986b) found a marginal but systematic difference between the L - sigma - Mg2 relations in the Virgo and Coma clusters. Also, Djorgovski et al. (1989) find that the Dn - sigma relation in different clusters, varies with cluster richness class. Within clusters, it varies with distance from the cluster center. Further investigation of the fundamental plane and its dependence on environment is desirable but will require large bodies of high-quality data.



3 Uncertainty about whether there is a discontinuity (Binggeli et al. 1984, Sandage et al. 1985a, Binggeli 1985, Caldwell & Bothun 1987) is based mainly on three problems. (a) Some global parameters used (e.g. isophotal mean surface brightnesses) are insensitive structure indicators. (b) Bright dSph galaxies, which are close to the E sequence, often contain both an exponential component and (apparently) a bulge (e.g. NGC 5206; Caldwell & Bothun 1987). Inclusion of bulges guarantees convergence with the E sequence; disks and bulges should be plotted separately in these diagrams. (c) Seeing effects are so large that it is very difficult to define the faint end of the E sequence using ground-based observations of galaxies as Far away as 20 Mpc (Kormendy 1987c). Effects (a)-(c) appear sufficient to explain the apparent merging of the E and dSph sequences seen by the above authors. Nevertheless, it is important that the galaxies they cite as transition objects be measured with Space Telescope for inclusion in the parameter diagrams. Back.

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