This section and the next discuss two kinds of features which are closely connected with the bar phenomenon, and which may be produced by it. Lens components are elliptical features of nearly constant surface brightness which are found between the bulge and the disk; both principal axes have shallow brightness gradients interior to a sharp outer edge, and then steep outer gradients. Figure 40 shows an example in NGC 1291. The steepness of the outer gradient is illustrated by the fact that the lens is only a little larger in the deeper photograph. Other lenses are illustrated in the Hubble Atlas, in Sandage and Brucato (1979) and in Kormendy (1979a). This last paper also gives a detailed discussion of lens morphology. Despite the fact that there is little published photometry of lenses, it is already clear that the strongest lenses contribute more of the light of a galaxy than the strongest bars (see examples in Kormendy 1977a). Although lenses occur in some unbarred galaxies (e.g., NGC 1553, Freeman 1975b; Sandage and Brucato 1979; NGC 2775, Hubble Atlas), they are especially common in barred galaxies. In fact, Kormendy (1979a) has found that ~ 54% of SB0-a galaxies have lenses. Furthermore, in almost every case the bar exactly fills the lens in one dimension (e.g., NGC 1291 in Fig. 40). This high degree of morphological regularity is trying to tell us something. In general, the evidence summarized by Kormendy (1979a) suggests that there is a close connection between lenses and bars.
Further examination of lens properties reveals other connections with bars. One example involves the shapes of lenses. The most secure determination is for NGC 1291. Mebold et al. (1979) derive a maximum H I rotation velocity of only 20 ± 5 km s-1 in this galaxy, showing that it is almost face-on. At the implied inclination of i = 84° ± 2°, a circular disk would have an apparent axial ratio of 0.995 ± 0.004. Therefore the axial ratio of the lens shown in Figure 40 is the true axial ratio, b/a = 0.81 ± 0.02, with the bar filling the longest axis. (It seems reasonable to assume that the visible galaxy is coplanar, since isolated edge-on galaxies do not generally show warps at such high surface brightnesses.) A statistical study of lens shapes based on the distribution of their apparent axial ratios is discussed by Athanassoula et al. (1982). This implies that lenses typically have axial ratios in the equatorial plane of b/a ~ 0.85 ± 0.1, with the bar filling the longest dimension. In the axial direction, lenses seem to be as flat as disks (e.g., Tsikoudi 1977, 1980; Burstein 1979d).
Kinematic properties of lenses are being studied by Kormendy (1982b). Figure 47 shows major-axis rotation and dispersion curves for two galaxies with especially well developed lenses; NGC 3945 and NGC 1553. The behavior of these galaxies is remarkably similar. In both cases the velocity dispersion decreases in the outer part of the bulge. Where the bulge fades into the lens (r ~ 17" in NGC 3945 and r ~ 15" in NGC 1553, see Freeman 1975b), the rotation velocity decreases and the dispersion increases. Thus the inner parts of these lenses are quite hot. At larger radii the dispersion decreases until at the edge of the lens it is too small to be measured reliably.
Figure 47. Major-axis rotation and velocity dispersion data in NGC 3945 [SB(lens)0] and in NGG 1553 [SA(lens)0]. The radius of the lens is 51" in NGC 3945 and 36" in NGC 1553. The instrumental velocity dispersion is ~ 100 km s-1 for NGC 3945 and ~ 65 km s-1 for NGC 1553.
The relative importance of rotation and heat can be estimated by plotting the local ratio of V / against radius (Fig. 48; cf. Fig. 45). This figure leads to the following conclusions (Kormendy 1981, 1982b). (1) Lenses in barred galaxies are as hot as n-body models of bars. Since these models have local V / values similar to those of real bars (section 5.1.5), the relative importance of rotation and heat is similar in lenses and bars. In both cases V/ ~ 0.8 at r ~ 0.4rl, so the ratio of rotational to random kinetic energy is ~ 1/2 in the inner parts of lenses. (2) Lenses in two unbarred galaxies are as hot as SB lenses. Therefore lenses are not hot only through the heating effects of the bar (e.g., Hohl 1971), unless all lenses contained bars in the past. (3) The rim of a lens is cold: is at least as small as the instrumental dispersion in almost all galaxies illustrated in Figure 48. This is not surprising, in view of the fact that lenses have sharp edges.
Figure 48. Local ratio of rotation velocity to azimuthal velocity dispersion in lenses of barred and unbarred galaxies. Radii are normalized to the radius of the lens for galaxies and to the corotation radius for the n-body models. Horizontal "error bars" show the range in radii over which data were averaged; the resulting point is plotted at the weighted mean radius of the individual measurements. Open symbols refer to measurements contaminated by bulge light. This figure is taken from Kormendy (1981).
The physical nature of the relationship between bars and lenses is not understood. Two suggestions have been made on how lenses might originate.
Based on their morphology, Kormendy (1979a) suggested that lenses might result from a process of secular evolution which makes some bars evolve away to a nearly axisymmetric state. Since half of all SB0-a galaxies contain both lenses and bars, any such process (1) must be able to switch itself off before bar dissolution is complete, or (2) must have a timescale comparable to a Hubble time, or (3) must begin relatively late in a galaxy's history. More recent results are consistent with the above suggestion. The subject is reviewed in Kormendy (1981); I give only a summary. The bar is assumed to interact with some other component in such a way that stars gradually escape from the bar. Since clear-cut lenses occur in early-type galaxies, the component interacting with the bar may be the bulge. Investigations of the interaction of bars and halos have concentrated on slowly rotating bulges (Sellwood 1980). In such cases angular momentum flows from the bar to the halo, and the bar generally gets stronger. However, SB bulges rotate very rapidly (Figures 41, 46, 47). Almost certainly they rotate rapidly enough so that there is an inner Lindblad resonance; i.e., so that the elongations of some bulge orbits precess faster than the bar. In this case angular momentum may flow from bulge to bar, making the bar less elongated. As the potential becomes more nearly circular, some stars may cease to precess at the bar rate and instead fill a nearly circular disk of the same size as the bar. The observation that bars and lenses are kinematically similar supports this suggestion. But we basically do not know how a bar would interact with a rapidly rotating bulge. An n-body experiment is needed.
A second possible way of forming lenses is being explored by Athanassoula and Sellwood (1982). They consider the effect on a global bar instability of varying the initial velocity dispersion in the disk. Preliminary results indicate that instabilities grow more slowly in hotter disks, and stabilize at rounder shapes. This is at least qualitatively consistent with the observation that the inner parts of lenses are hot. The questions still to be answered center on whether or not a realistic lens results. (1) Do the ovals which form in hot disks have sharp edges? (2) How do a bar and a lens of the same size form in so many galaxies? Finally, (3) how does one plausibly create an initial condition in which a stellar disk is very hot in two directions but cold in the third?
The above are two possible ways of forming lenses with properties similar to bars. Both require further work. Both involve n-body questions which, although they are difficult, are well-defined.