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4.2.6. The Dynamical Importance of Rotation in Bulges and Ellipticals

Until 1975 it was believed that elliptical galaxies are simple dynamical systems which are flattened by rotation but which otherwise are like globular clusters or the isotropic King (1966) models (Prendergast and Tomer 1970; Wilson 1975). Soon thereafter it became clear that they are generally triaxial systems with little significant rotation (see Illingworth 1981 for a review). Such objects owe their shapes to anisotropic velocity dispersions (see Binney 1978b, 1981b, 1982a, b for reviews). I will discuss this story briefly, and will then concentrate on recent developments.

The first indication that rotation is unimportant in some ellipticals came when Bertola and Capaccioli (1975) published a rotation curve of NGC 4697. They observed a maximum projected rotation velocity of ~ 60 km s-1, corresponding to a true rotation of ~ 85 km s-1. This is remarkably small compared to the central velocity dispersion sigma = 310 km s-1 derived by King and Minkowski (1966). Binney (1976) notes that the implied ratio of kinetic energies in rotation and random motions is not enough to maintain the E5 shape of the galaxy. Ironically, a more accurate dispersion measurement and a rotation curve which reaches larger radii (Fig. 36) show that NGC 4697 is actually one of the few rapidly rotating ellipticals (Illingworth 1981). However, the conceptual dam had been burst. Shortly thereafter, Illingworth (1977) demonstrated that rotation is too small to account for the observed flattening in 12 of 13 galaxies studied.

An explanation appeared almost simultaneously, as Binney (1976) realized that protogalactic velocity anisotropies were likely to survive the collapse and lead to ellipticals with anisotropic dispersion tensors. Rotation is then not necessary to produce the flattening. Indeed, there is no reason why any two of the principal dispersion components should be equal, so ellipticals might well be triaxial. Despite the fact that this picture was a conceptual revolution, it became established very quickly, because of the concurrent kinematic measurements, and because triaxial ellipticals provided a natural explanation for the known but neglected observation of isophote twists (section 3.3.1). A great deal of activity was stimulated, including further searches for isophote twists (section 3.3.1), rotation measurements (Peterson 1978b; Sargent et al. 1978; Young et al. 1978b; Schechter and Gunn 1978; Dressler 1979; Jenkins and Scheuer 1980; Efstathiou, Ellis and Carter 1980, 1982; Davies 1981; Carter et al. 1981; Williams 1981; Davies et al. 1983; Davies and Illingworth 1982b; Fried and Illingworth 1982) and theoretical studies (Binney 1978a, b; 1980a, b; 1981b; 1982a, b; Fall 1982).

The results are illustrated in Figures 37, 38a and 46. These are the usual Vm / sigma bar - varepsilon diagrams (Illingworth 1977), which compare the global dynamical importance of rotation and random motions. Recall that Vm / sigma bar measures the fraction of the dynamical support which is provided by rotation: (Vm / alpha sigma bar)2 is the ratio of rotational to random kinetic energy, where 21/2 leq alpha leq 31/2, depending on the amount of anisotropy.

Consider first Figure 37, which shows the now-classical results for bright ellipticals (-19.5 geq MB geq - 23.5). The predicted rotation for isotropic oblate spheroids is shown by the "oblate line", labeled ISO in the figure. The theoretical parameters are the mass-weighted mean velocity and dispersion, corrected for projection effects, for edge-on models of constant ellipticity varepsilon. The equation of the oblate line is implicit in Binney's derivations; the following explicit statement is kindly provided by Fall (1981):

Figure 37

Figure 37. Comparison of the dynamical importance of rotation in ellipticals (crosses), bulges (circles) and various dynamical models (curves, from Binney 1978a, 1980a). Vm is the maximum projected rotation velocity; sigma bar is the projected dispersion between r ~ 2" and r ~ re/2. The ellipticity is varepsilon = 1 - b/a, b/a the axial ratio. The line labeled ISO represents projected models of oblate spheroids with isotropic residual velocities and rotational flattening. The line labeled ANISO represents typical anisotropic oblate models with axial dispersion sigmaz smaller than sigmar and sigmatheta. This figure is taken from Kormendy and Illingworth (1982a).

Equation 26 (26)

Here e = (1 - b2 / a2)1/2 is the eccentricity, varepsilon = 1 - b/a is the ellipticity, and b/a is the axial ratio. For back-of-the-envelope calculations, it is useful to know that the approximation,

Equation 27 (27)

is correct to (i.e., too large by) 1 ± 1/2% in the entire range 0 leq varepsilon leq 0.95 (E0 - E9.5). At small varepsilon, equation (27) is the parabola V bar / sigma bar = varepsilon1/2 referred to by Binney (1982a). The oblate line is calculated for edge-on models. Fortunately, a change in the viewing inclination moves a model downward along the oblate line if the intrinsic ellipticity varepsilon ltapprox 0.6. This is illustrated in Figure 1 of Illingworth (1977), in which a number of specific models viewed at various inclinations all fall close to the oblate line. Thus, oblate rotators should lie along the ISO line in Figure 37 with little scatter.

Before any comparisons with the observations are made, it is important to note the following complications. First, the published models show a significant amount of scatter about the oblate line (Illingworth 1977). Second, there is no exact observable analogue of the mass-weighted V bar and sigma bar of equation (26). Binney (1980a) has shown that Vm / sigma0, sigma0 the central dispersion, approximates V bar / sigma bar, although not perfectly. Furthermore, it is clear that galaxies are more complicated than the models. In particular, varepsilon usually varies with radius, and the maximum rotation and maximum flattening sometimes occur at very different radii. The adopted values Vm / sigma bar and an average varepsilon biased toward the maximum value are compromises whose derivation is discussed in Binney (1980a, 1982a), Kormendy and Illingworth (1982a) and Kormendy (1982a). These complications are not critical as long as any differences between galaxy samples are large and clearcut, but they become important when more precise comparisons with the models are made (e.g., section 5.2).

Figure 37 shows that ellipticals brighter than MB appeq -20.5 rotate only 1/3 - 2/3 as rapidly as models of isotropic oblate spheroids. This means that they have only 1/9 - 1/2 as much rotational energy as they would need if they were flattened only by rotation. What kind of model does this low rotation require? One possibility which appears to fail is a population of isotropic prolate spheroids which has the observed distribution of flattening. Binney (1978a, Fig. 2) has derived the distribution of such models in the Vm / sigma bar - varepsilon diagram. The scatter at a given varepsilon is large. For example, a prolate spheroid seen broadside-on has the same apparent varepsilon when viewed from its rotation axis as it does in its equatorial plane, but Vm is large in the latter case and zero in the former. It is convenient to represent a distribution of prolata by a median line in the Vm / sigma bar - varepsilon diagram, such that half of the models are expected to fall above the line and half below it. Such a median line from Binney (1978a) is shown in Figure 46. These models still rotate more rapidly than the galaxies, although the difference is not large. More consistent with the observations are models with some dispersion anisotropy (e.g., the ANISO line in Fig 37, from Binney 1978a, Fig. 1). In particular, if the axial velocity dispersion sigmaz is made smaller than the other two components sigmar and sigmatheta, then relatively little rotation is required to account for even large amounts of flattening. Nearly all of the observations can be explained if 1 - delta appeq sigmaz2 / sigmar2 is slightly less than 1, say 0.7 < 1 - delta < 1.0 (Binney 1981b, Fig. 1a). Of course, a variety of triaxial forms is also possible. That some ellipticals are triaxial is confirmed by observations of isophote twists (section 3.3.1) and minor-axis rotation (Schechter and Gunn 1978; Jenkins and Scheuer 1980; Williams 1981; Bertola 1981).

About 15% of bright ellipticals rotate essentially as rapidly as the oblate models (Illingworth 1981). Two examples are NGC 3557 and NGC 4697 (Illingworth 1977, 1981; Davies 1981).

Rotation rates have also been measured in a number of bulges (Illingworth and Schechter 1982; Kormendy and Illingworth 1982a; Kormendy 1982a; and references therein). As shown in Figure 37, the bulges of unbarred galaxies rotate much more rapidly than most bright ellipticals. In fact, they are essentially consistent with the oblate line, especially since they are flattened by the disk potential (Monet, Richstone and Schechter 1981) as well as by rotation. Any anisotropy is very small. Neglecting the disk potential, the bulge data are formally consistent with sigmaz / sigmar appeq 0.97. The bulges are more uniform in their rotation properties than ellipticals, despite the large range in Hubble types (S0 - b) and bulge-to-disk ratios (0.25 leq B/D leq 11). Note that the sample in Figure 37 is biased toward small varepsilon and large B/D (Kormendy and Illingworth 1982a). Thus rotation very likely dominates the dynamics of most bulges. Data on bulges of barred galaxies confirm this conclusion (section 5.2, Fig. 46).

Figure 38

Figure 38. (a, upper) Vm / sigma bar - varepsilon diagram analogous to Fig. 37, but adding ellipticals fainter than MB = - 20.5 as filled circles. Brighter ellipticals are shown as open circles. SA bulges (crosses) and the oblate line are reproduced from Fig. 37. (b, lower). For bulges (crosses) and ellipticals (filled circles), (V / sigma)* is plotted against absolute magnitude. (V / sigma)* is the ratio of the observed value of Vm / sigma bar to the value predicted by the oblate line for the given ellipticity. These figures are taken from Davies et al. (1983).

We cannot conclude from the above observations that bulges and ellipticals are very different. Schechter (1981) emphasizes that the ellipticals in Figure 37 (-19.5 geq MB geq -23.5) are, on average, brighter than the bulges ( -18.5 geq MB geq -21). Only NGC 4594 has a bulge as bright as the typical elliptical studied (MB appeq -22.2). Therefore, Davies et al. (1983) have studied the rotation properties of ellipticals fainter than MB = -20.8, to allow a more direct comparison with bulges. Also, they aimed to study the importance of rotation as a function of absolute magnitude, to help to discriminate between various theories of the origin of angular momentum. The results are shown in Figure 38. There is a clear correlation of rotation properties with absolute magnitude. We can measure the importance of rotation with a parameter (V / sigma)* = (Vm / sigma bar)obs / (V bar / sigma bar)oblate, the ratio of the observed value to that predicted by the oblate line. A plot of (V / sigma)* vs. absolute magnitude shows a clear trend. Faint ellipticals rotate as rapidly as bulges of comparable luminosity; bright ellipticals have a large spread in rotation measures but generally rotate slowly. It is not clear whether (V / sigma)* values for bulges also correlate with MB, because only one bright bulge has been measured in detail (NGC 4594). To clarify any differences between bulges and ellipticals we need to observe more bright bulges. This will not be easy, because bulges as luminous as the one in NGC 4594 are rare and difficult to identify unless they are nearly edge-on. However, a recent study by Dressler and Sandage (1983) suggests that very luminous bulges are generally rapid rotators.

Davies and collaborators go on to investigate the implications of the (V / sigma)* - MB correlation for theories of galaxy formation. Their discussion is beyond the scope of this paper. The main conclusion is that the shape and especially the amplitude of the (V / sigma)* - MB correlation are too large to be produced by tidal torques during dissipationless collapse in a hierarchical clustering picture. The data are more consistent with dissipational galaxy formation. However, a wide variety of formation processes is possible. The various theories are not well enough developed to make predictions that can be tested conclusively.

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