Introduction:

Some 30% of disc galaxies have a pronounced central bar feature in the disc plane and many more have weaker features of a similar kind. Kinematic data indicate that the bar constitutes a major non-axisymmetric component of the mass distribution and that the bar pattern tumbles rapidly about the axis normal to the disc plane. The observed motions are consistent with material within the bar streaming along highly elongated orbits aligned with the rotating major axis. A barred galaxy may also contain a spheroidal bulge at its center, spirals in the outer disc and, less commonly, other features such as a ring or lens. Mild asymmetries in both the light and kinematics are quite common.

We review the main problems presented by these complicated dynamical systems and summarize the effort so far made towards their solution, emphasizing results which appear secure. Bars are probably formed through a global dynamical instability of a rotationally supported galactic disc. Studies of the orbital structure seem to indicate that most stars in the bar follow regular orbits but that a small fraction may be stochastic. Theoretical work on the three-dimensional structure of bars is in its infancy, but first results suggest that bars should be thicker in the third dimension than the disc from which they formed. Gas flow patterns within bars seem to be reasonably well understood, as are the conditions under which straight offset dust lanes are formed. However, no observation so far supports the widely held idea that the spiral arms are the driven response to the bar, while evidence accumulates that the spiral patterns are distinct dynamical features having a different pattern speed. Both the gaseous and stellar distributions are expected to evolve on a time-scale of many bar rotation periods.

Galaxies are beautiful objects, and the graceful symmetry of many barred galaxies is particularly striking. One of the best examples close enough for us to examine in detail is NGC 1365 in the Fornax cluster. Such objects are doubly pleasing to the astro-physicist because they also present a number of very challenging dynamical problems.

Barred galaxies are a heterogeneous class of objects. The bar component ranges from a minor non-axisymmetric perturbation to a major feature in the light distribution. Other properties, such as the size of the bar relative to the host galaxy, the degree of overall symmetry, the existence of rings and the numbers (and position relative to the bar) of spiral arms in the outer disc, the gas and dust content, etc.,vary considerably from galaxy to galaxy.

Bars can be found in all types of disc galaxies, from the earliest to the latest stages of the Hubble sequence. Because there is a continuum of apparent bar strengths from very weak oval distortions to major features, it becomes more a matter of taste to decide what strength of bar in a galaxy is sufficient to merit a barred classification. Apart from the very latest types, there is rough agreement over the fractions containing strong bars; when combined over all stages, the SB family constitutes between 25% and 35% of all disc galaxies. However, there is considerably less agreement over intermediate cases, the SAB family: the morphological classifications assigned in the RC2 indicate a combined fraction for the SAB family as high as 26.4% – substantially higher than in either of the other two catalogs. Notwithstanding these variations, it is clear that barred galaxies constitute a major fraction of all disc galaxies.

A small number of galaxies which appear unbarred at visual wavelengths have been found to be barred when observed in the near infra-red. The three clearest cases are NGC 1566 (Hackwell and Schweizer 1983), NGC 1068 (Scoville et al. 1988, Thronson et al. 1989) and NGC 309 (Block and Wainscoat 1991); much shorter or weaker bar-like features have shown up in many others . Many more such cases could yet be discovered, as IR cameras are in their infancy; if so, the fraction of barred galaxies could turn out to be much higher than indicated.

Elliptical galaxies are sometimes described as stellar bars, which is apt for certain purposes. However, we exclude such galaxies from this review because, unlike barred galaxies, they appear to be single component systems with no surrounding disc or distinct central bulge. Moreover, the stellar distribution is not thought to be as flattened, or to rotate as fast as bars in disc systems. De Zeeuw and Franx (1991) have surveyed the literature on the dynamics of these objects.

We are still far from a complete understanding of the dynamical structure of barred galaxies. We would really like to know the three-dimensional density distribution of each galaxy in order to be able to calculate the gravitational potential. This information, combined with the rotation rate of the bar, would enable us to calculate the motion of stars and gas. We could claim we understood the dynamical equilibrium if we were able to propose a self-consistent model in which the various orbits were populated so as to reconstruct the observed mass distribution. We should then ask whether the equilibrium model would be dynamically stable, or how it would evolve, how the object was formed in the first place, etc. Here we will be able to do no more than scratch the surface of the majority of these problems.

There are many excuses for our ignorance:

i. The distance of these objects, combined with a surface brightness which declines to below that of the night sky, makes it very hard to make measurements of the quality required.

ii. We see these objects only in projection. As we believe they are disc-like, we might na¨ıvely hope that there is a plane of symmetry, but because barred galaxies are intrinsically non-circular, determination of the inclination angle is more difficult than in their unbarred counterparts.

iii. Worse still, it is unlikely that a single plane of symmetry exists in many galaxies. Those seen edge-on frequently exhibit warps mainly in the outer parts, yet this is precisely where the problem of point (2) is best avoided.

iv. We have very little knowledge of the thickness along the line of sight; it is reasonable to believe that the thickness of discs seen edge-on is typical, but some edge-on systems have box-shaped bulges – we have no way of telling whether these are strongly barred galaxies, or whether the box shape is uncorrelated with the morphology in the plane.

v. The line-of-sight velocity component, which is all we can measure, also gives us no more than the average integrated through the object, or to the point at which it becomes opaque (itself a controversial question; e.g. Disney et al. 1989, Valentijn 1990, Huizinga and van Albada 1992, etc.).

vi. Our “snapshot” view of each galaxy prevents our making direct measurements of the rotation rate (or pattern speed) of the bar; the angular speed of the non-axisymmetric features inferred from the observed motions of gas and stars are highly uncertain.

vii. Many features, e.g. dust lanes and rings, result from dissipative processes in the gaseous component. Unfortunately, we lack a good theoretical description for the large-scale behavior of gas, which is stirred and damped on scales of a few parsecs, while we wish to model gaseous features traceable over many kiloparsecs.

viii. We are uncertain of the extent to which the internal dynamics are affected by dark matter. At large radii, the orbital motion in barred galaxies resembles the nearly circular rotation pattern typical of a normal spiral galaxy and, as emphasized by Bosma (1992), the mass again appears to have a different distribution from the light. There is some evidence that the luminous matter dominates the dynamics of the inner parts of unbarred galaxies (e.g. Casertano and van Albada, 1990) and we shall proceed by assuming that this is the case in the bar region. This, perhaps rash, assumption is buttressed by the kinematic evidence that the more prominent bars are associated with large departures from simple circular motion in the disc plane, indicating that the bar contains a significant fraction of the mass in the inner galaxy.

ix. The weakest excuse is that our mathematical ability is so limited that we are unable to solve the equations which should describe the structure of such objects, except in a few highly idealized cases. Our understanding of the dynamics has therefore to be laboriously pieced together using numerical techniques, which themselves have limitations. It is customary to break the light of a barred galaxy into a number of different components, or building blocks: a disc, a bulge, a bar and, sometimes, a lens and/or rings (e.g. Kormendy 1979). Most of these components are evident, only the lens requires description: if present, it is a comparatively bright, oval part of the inner disc surrounding the bar. It is distinguished by a moderately sharp edge, which causes a locally steep gradient in the photometric profile. Although the working hypothesis of separate components seems justified by the fact that the individual features are readily distinguished by the eye, it is much more difficult to separate the components quantitatively. The idea of morphologically distinct components has been taken much further, however, and each component is frequently assumed to be a separate dynamical entity. This fundamental assumption is rarely discussed or even stated. Because it is obviously much easier to understand the dynamics of each apparent component separately, we also treat them as dynamically independent for the first part of ther eview, and examine the validity of this assumption only towards the end.

Most of the mass in the inner bright regions of a galaxy is in stars; the gas mass is rarely sufficient to affect the gravitational potential and we have already indicated that we assume dark matter begins to dominate only in the faint outer parts. We therefore consider the dynamics of the inner galaxy to be that of a purely stellar system in which any gas present acts as tracer material. Galaxies are thought to be of intermediate dynamical age; a typical star might have completed some 50 orbits around the center. This is neither so old that the galaxy must be in settled equilibrium (the existence of spiral structure and on-going star formation show this is not the case) nor so young that its morphological features just reflect initial transients. It is most appropriate, therefore, to consider the slow evolution of a nearly equilibrium model. Accordingly we discuss equilibrium models before we go on to consider how they might have arisen and will evolve. We conclude by considering interactions between the different components.

Observed properties of bars

We begin by summarizing the observational information relating to the dynamical structure of the stellar bar, and leave data on gas to and on rings to, where we discuss these other phenomena. Unfortunately we cannot make direct measurements of even the most basic properties, and are forced to make indirect inferences from the observables. Our requirements fall into three general areas:

i The distribution of mass, in order to determine the gravitational potential. This has to be deduced from measurements of (a) the light distributions and (b) the velocity field.

ii The bar rotation velocity, or pattern speed, which can be inferred only with considerable uncertainty from the velocity field or, still less directly, through modelling the gas flow pattern.

iii The full three-dimensional velocity dispersion which, jointly with orbital streaming, determines the stellar dynamical equilibrium. This can be constructed from the projected velocity dispersion data only with the aid of a mass model.

Components of the light distribution

Deprojection of the light distribution of barred galaxies is more difficult than for nearly axis-symmetric galaxies since it is far from clear that any isophote should be intrinsically round. The inclination is generally inferred by assuming that the outer faint isophotes are projected circles, but it should be borne in mind that even far out the shape could be intrinsically elliptical, especially if an outer ring is present, or the plane is warped.

The major axis of the bar is always less than the diameter of the host galaxy. In general, bars in late-type systems are shorter relative to the total galaxy size (Dbar/D25 = 0.2 to 0.3) than those in early-type galaxies (Dbar/D25 = 0.3 to 0.6), where D25 is the diameter at which the surface brightness of the galaxy falls below 25 mag arcsec−2 (e.g. Athanassoula and Martinet 1980, Elmegreen and Elmegreen 1985, Duval and Monnet 1985).

There also appears to be a correlation between the length of the bar and the size of the bulge. Athanassoula and Martinet (1980) suggest that the deprojected length of the bar scales as 2.34 times the bulge diameter, while Baumgart and Peterson (1986) estimate 2.6 ± 0.7.

2.2. Fraction of total luminosity in the bar estimates of the luminosity fraction in the bar depend not only on the radius to which the total light is measured but also on how the decomposition is performed. Luminosity fractions are usually quoted as a fraction of either the light out to the end of the bar, or integrated out to some faint isophotal level, e.g. D25.

Decompositions were originally performed by attempting to fit the components with idealized models: prolate bars, exponential discs, etc. (e.g. Crane 1975, Okamura 1978, Duval and Athanassoula 1983, Duval and Monnet 1985). Blackman (1983) used breaks, or changes of slope, in the photometric profile to define the boundaries between the different components. Others have tried Fourier analysis of the light distribution (e.g. Elmegreen and Elmegreen 1985, Buta 1986b, 1987, Ohta et al. 1990, Athanassoula and Wozniak 1992), which imposes no prejudice as to the form of the components and gives direct measurements of the strengths of the different non-axisymmetric features, once the inclination is determined. Probably the most successful decomposition technique for a barred galaxy was proposed by Kent and Glaudell (1989) who devised an iterative method to separate an oblate bulge model from the bar.

It is not surprising therefore that the estimates for the same object vary widely from author to author. For example, for the SBc galaxy NGC 7479 they range from the unrealistically low value of 8% by the modelling technique (Duval and Monnet 1985) to a more likely 40% by direct estimation of the light within an approximate isophote at the edge of the bar (Blackman 1983) and 38% by Fourier decomposition (Elmegreen and Elmegreen 1985). Ohta et al. (1990) give values in the range 25 to 50% in the “bar region” to 10 to 30% out to D25 for their sample of early-type galaxies.

The light distribution within the bar

Generally, the shapes, strengths and lengths of bars seem to vary systematically from early to late-type systems.

Early-type galaxies. Many bars in galaxies of types SB0 and SBa have a pronounced rectangular shape seen in projection (Ohta et al. 1990, Athanassoula et al. 1990) with axial ratios between 0.3 to 0.1. The surface brightness decreases slowly along the bar major axis, in some cases as a shallow exponential but in others it is almost constant until close to the end of the bar (e.g. Elmegreen and Elmegreen 1985, Kent and Glaudell 1989). There is no difference between the leading and trailing sides (Ohta et al.)

The surface brightness contrast between bars and the axis-symmetric component can range from 2.5 to 5.5 (Ohta et al.). Bars are considerably brighter on the major axis, and usually fainter on the minor axis, than the inwardly extrapolated disc profile. Azimuthally averaged, however, the radial profiles are no more varied than those of ordinary spirals; Wozniak and Pierce (1991) fit an exponential to the disc profile outside the bar but ring features, which are much more common in barred galaxies, can make this a poor approximation (e.g. Buta 1986b).

Late-type galaxies. The light distribution in late-type galaxies is generally less smooth owing to greater dust obscuration and more intense knots of young stars. The difficulties created by both these problems are lessened by making the observations at wavelengths as far into the infra-red as possible (e.g. Adamson et al. 1987).

Bars in galaxies of types SBbc to SBm are generally more elliptical (e.g. Duval and Monnet 1985), shorter and weaker than their earlier counterparts. They also begin to show quite pronounced asymmetries, e.g. one end may appear squarer than the other. The surface brightness distribution of weak bars in late-type systems is much more centrally peaked and falls off exponentially along the bar, sometimes even more steeply than the disk. Across the bar, the light profile is close to Gaussian (e.g. Blackman 1983).

Relaxation Time

Since the density of stars throughout the main body of a galaxy is very low, the orbit of an individual star is governed by the large-scale gravitational field of the galaxy and is not appreciably affected by the attraction of the relatively few nearby stars. The gravitational impulses received by a star as it moves through a random distribution of scattering stars nevertheless accumulate over time. The relaxation time is the time taken for these cumulative random deflections to change the velocity components along the orbit of a typical star by an amount equal to the stellar velocity dispersion.

Chandrasekhar’s (1941) formula yields a value of 10exp13 years, or 1000 times the age of the universe, for star-star encounters in the neighborhood of the Sun. Since the relaxation time varies inversely as the stellar density, considerably shorter timescales apply in the centers of galaxies, where the star density is higher by several orders of magnitude.

Chandrasekhar’s formula has not required revision, but the impressively long relaxation time it predicts is believed to be an overestimate. We now know that the distribution of mass in disc galaxies is not always as smooth as he assumed – gas is accumulated into giant molecular cloud complexes, ranging in mass up to 105 Msun or even 106 Msun, and some bound star clusters are known to contain almost as much mass in stars. Although these objects are much more diffuse than point masses, clumps in this mass range shorten the relaxation time considerably.

It is even further reduced by the tendency for the gravitational attraction of massive objects to raise the stellar density near themselves. The accumulated material, known as a polarization cloud, can easily exceed the mass of the perturber when the stellar motions are highly ordered – as in a rotationally supported disc (Julian and Toomre 1966). Nevertheless, the relaxation time in the objects we consider here is never shorter than the orbital period, and is usually considerably longer.

It should be clear that we are still far from a complete understanding of the basic structure of barred galaxies, but progress has been rapid in recent years – especially where orbit studies have been connected to observational or simulated results. There are a few aspects where the observed facts seem to fit reasonably well with theoretical ideas.

On the bright side, it seems very likely that strong bars are formed by the dynamical instability discussed in. Bars formed in the N-body experiments end near co-rotation and have mass distributions and kinematic properties which seem to correspond with those observed, though more detailed comparisons would be desirable. Moreover, such bar models appear to have the right strength, and to rotate sufficiently rapidly to shock gas in places resembling the dust lane patterns in some barred galaxies. The flow patterns described in are understood in terms of orbit theory and fit the observed kinematics for entirely plausible model parameters. We also think we understand the origins of rings sufficiently well to be able to interpret them as signatures of major resonances with the bar pattern.

These successes add up to a compelling, but indirect, case for a bar pattern speed in galaxies which places the major axis Lagrange point just beyond the bar’s end. Such a value is consistent with the more direct observational estimates of pattern speed which unfortunately are subject to large uncertainties.

Current observational work is beginning to provide more quantitative data on the light distributions and kinematics of barred galaxies. In particular, the kind of comparison with theoretical models made by Kent and Glaudell (1989) for NGC 936 should be extended to several other galaxies in case the structure of that galaxy is special in any way. Moreover, the theory of ring formation is in dire need of more detailed observational comparisons; we need good kinematic maps of many more such galaxies to confirm that the rings do indeed lie at the resonances for the bar.

Yet our understanding of the dynamical structure of bars is still far from complete. The vast literature on orbits in bar-like potentials has led to a few major conclusions relevant to the structure of barred galaxies: the most important is that the majority of stars within the bar probably follow eccentric orbits which are trapped, or semi-trapped, about the main family of orbits aligned with the bar (the x1 family). Apart from this, knowledge of the main orbit families has improved our understanding of gas flows and indicates that it would be hopeless to try to construct a self-consistent bar having a mean rotational streaming in a sense counter to the pattern rotation. It is progress, of a kind, to learn that a simple generalization of the two-dimensional orbital structure does not provide a complete description of three-dimensional bars. Only one set of approximately self-consistent two-dimensional numerical solutions has been found (Pfenniger 1984b) and the prospects for analytic models in the near future are bleak. Next to this, our fragmentary understanding of the evolution of barred galaxies, and almost total blanks on the origins of lenses and the pronounced asymmetries in some galaxies, seem of secondary concern.

Probably the most pressing need on the theoretical side is for a more sustained attack on the orbital structure of three-dimensional bars, preferably through studies of rapidly rotating three-dimensional objects having density distributions resembling those seen in galaxies. It should be possible to address the evolutionary issues as more powerful computers enable the quality of N-body simulations to rise.

Other fundamental questions also need to be pursued. The prime candidate is what determines the 30% fraction of galaxies seen to have strong bars. Since the bar instability seems able to create strong bars in nearly all disc galaxies, how can we account for the current moderate fraction? The ideas for controlling the bar instability and those for destroying bars do not explain either why some 70% of galaxies manage to avoid a bar instability, or why the bar was subsequently weakened or destroyed in that fraction of galaxies. Are the weak bars in galaxies of intermediate type (SAB) formed through the partial dissolution of strong bars, or is some totally different mechanism required?

The bar instability assumed an initially unstable equilibrium disc without specifying how that could have been created. A discussion of the formation of disc galaxies is well beyond the scope of this review, but it does seem likely that the bar instability would be profoundly affected by the manner in which galactic discs form. Sellwood and Carlberg (1984), in a few preliminary experiments mimicking gradual disc formation, found that the velocity dispersion of the stars rose sufficiently rapidly to inhibit the formation of a strong bar as the disc mass built up. Further experiments of this kind, especially including gas dynamics seem warranted.

Another issue is whether it is sensible to separate a barred galaxy into distinct dynamical components. Most theoretical and observational work has proceeded on the assumption that the bar and the bulge are distinct, but we have seen in that there may be no dynamical basis for considering them as separable. Moreover, as it seems likely that the bar formed from the disc, we may confuse ourselves by trying to understand these also as unrelated components.

Although we have a lot more to do before we can claim to understand the structure of these objects, we should be encouraged that progress over the last few years has been rapid, especially since the knowledge which enabled us to formulate many of these questions has only recently been acquired. The many new telescopes and observational techniques, particularly operating in the infra-red, are likely to advance the subject at its currently intense rate.

Gas and Dust

Although a small fraction of the total mass, at least in early type galaxies, the gas component is of considerable interest to the dynamicist mainly because it is an excellent tracer material. We have much more detailed knowledge of the flow patterns of gas in galaxies than we do of the stars because the Doppler shifts of the emission lines from excited gas are easier to measure than for the broader, weaker absorption lines seen in the composite spectra of a stellar population. Comparison between the observed flow pattern and the calculated gas behavior in a number of realistic potentials can be used as a means to estimate such uncertain quantities as the pattern speed and mass-to-light ratio of the bar.

The dust lanes, which are dark narrow features along spiral arms and bars where the gas and dust density may be several times higher than normal, also demand an explanation. The widely accepted view that these delineate shocks in the inter-stellar gas seems to have been first proposed by Prendergast (unpublished c1962).

Finally, the specific angular momentum of gas elements changes with time, causing significant radial flows of material. These are important for evolution of the metal content of the galaxy and can cause a build-up of gaseous material in rings where the flow stops.

Almost all the theoretical work and simulations have neglected motion in the third dimension. This approximation may still be adequate, notwithstanding the existence of transient bending instabilities and vertical instability strips within the bar since dissipation must ensure that the gas clouds remain in a thin layer. However, the work of Pfenniger and Norman(1990) may indicate that the radial flow of gas is accelerated as the material passes through vertically unstable regions.

Almost all the theoretical work and simulations have neglected motion in the third dimension. This approximation may still be adequate, notwithstanding the existence of transient bending instabilities and vertical instability strips within the bar since dissipation must ensure that the gas clouds remain in a thin layer. However, the work of Pfenniger and Norman (1990) may indicate that the radial flow of gas is accelerated as the material passes through vertically unstable regions.

Observations of gas in barred galaxies

Most data come from optical or 21cm HI observations and much less is known of the distribution and kinematics of the possibly dominant molecular gas component. This is because molecular hydrogen must be traced indirectly through mm-wave emission of CO and other species; the resolution of single dish antennae is low and only small portions of galaxies can be mapped with the current mm interferometers; for a recent review of available CO data see Combes (1992).

Gas distribution

The distribution of neutral hydrogen within each galaxy shows considerable variation. Neutral hydrogen appears to be deficient within the bar in a number of galaxies: e.g. NGC 1365 (Ondrechen and van der Hulst 1989) and NGC 3992 (Gottesman et al. 1984). On the other hand, counter-examples with significant HI in the bar are NGC 5383 (Sancisi et al. 1979), NGC 3359 (Ball 1986), NGC 4731 (Gottesman et al. 1984), NGC 1073 (England et al. 1990) and NGC 1097 (Ondrechen et al. 1989). The CO is sometimes distributed in a ring around the bar (e.g. Planesas et al. 1991) and sometimes concentrated towards the nucleus (e.g. Sandqvist et al. 1988).

Early type barred galaxies contain little gas, in common with their unbarred counterparts (e.g. Eder et al. 1991). Moreover, van Driel et al. (1988), who had two strongly barred galaxies (NGC 1291 and NGC 5101) in their sample having sufficient HI to be mapped, found that in both cases the gas was concentrated in an outer ring.

Kinematics

The position-velocity maps of gas in a barred galaxy indicate that the flow pattern is more complicated than the simple circular streaming (sometimes) seen in an approximately axisymmetric galaxy. In general, systematic variations in the observed velocity field produce characteristic S-shaped velocity contours and non-zero velocities on the minor axis, which are indicative of radial streaming. However, these features of the velocity field are seen only in those galaxies for which the viewing geometry is favourable, as emphasized by Pence and Blackman (1984b). The general morphology of the pattern is consistent with the gas following elliptical streamlines within the bar, but high resolution data sometimes show very abrupt changes in the observed velocity across a dust lane.

Optical and radio observations of the same galaxy are generally complementary. Although the HI gas is quite widely distributed, data from the high resolution aperture synthesis arrays has to be smoothed to a large beam (to improve the signal to noise) which blurs the maps, particularly near the centre where the velocity gradients are steep. Higher spatial resolution optical measurements of excited gas in the bright inner parts can overcome this inadequacy to some extent, especially from Fabry-Perot interferograms (e.g. Buta 1986b, Schommer et al. 1988, Duval et al. 1991), but strong optical emission tends to be very patchy, and is rarely found near the bar minor axis. mm data on molecular gas is also helpful in localized regions (e.g. Handa et al. 1990, Lord and Kenney 1991).

An excellent example is NGC 5383, one of the best studied galaxies; the Westerbork data of Sancisi et al. (1979) taken together with the optical slit data from Peterson et al. (1978), later supplemented by Duval and Athanassoula (1983), provided the sole challenge to theoretical models for many years. Fortunately, HI data from the Very Large Array (VLA) has become available in recent years, and the number of barred galaxies with well determined velocity fields is rising, albeit slowly.

In the majority of galaxies, the gas rotates in the same sense as the stars. However, exceptions have been found: NGC 2217 (Bettoni et al. 1990) and NGC 4546 (Bettoni et al.1991),

in which the gas in the plane can be seen to rotate in a sense counter to that of the stars. This most surprising aspect strongly suggests an external origin for the gas in these twoearly type galaxies and such cases are believed to be rare.

6.1.3. Dust lanes. Dust lanes occur more commonly in types SBb and later. Those along the bar are offset from the major axis towards the leading side (assuming the spiral to be trailing). Athanassoula (1984) distinguished two types: straight, lying at an angle to the bar as in NGC 1300, or curved as in NGC 6782 and NGC 1433. Sometimes predominantly straight lanes curve around the center to form a circum-nuclear ring. Dust can also be distributed in arcs and patches across the bar: NGC 1365 is a good example.

Evidence for shocks

Direct observations of steep velocity gradients across dust lanes, which would be the most compelling reason to believe these are shocks, have been hard to obtain. The two best examples are for NGC 6221 (Pence and Blackman 1984a) and for NGC 1365 (Lindblad and J¨ors¨ater 1987).

Evidence for gas compression also comes from the distribution of molecular gas, through CO emission, which appears to be concentrated in dust lanes (e.g. Handa et al. 1990, but see also Lord and Kenney 1991). Less direct evidence comes from the non-thermal radio continuum emission which is frequently strongly peaked along the dust lanes (e.g. Ondrechen 1985, Hummel et al. 1987a, Tilanus 1990) – the enhanced emission is consistent with gas compression, but could also have other causes.

Star formation and other activity

It has been noted frequently (e.g. Tubbs 1982, and references therein) that the distribution of young stars and HII regions is not uniform in barred galaxies. Stars appear to be forming prolifically near the centres (Hawarden et al. 1986, Sandqvist et al. 1988, Hummel et al. 1990) and at the ends of the bar, but not at intermediate points along the bar. This situation in some galaxies is so extreme as to have been interpreted as a star forming burst either in the nucleus or at the ends of the bar, e.g. NGC 4321 (Arsenault et al. 1988, Arsenault 1989). Dense concentrations of molecular gas are also sometimes found near the centers of barred galaxies (e.g. Gerin et al. 1988). It has also been noted by several authors (e.g. Simkin et al. 1980, Arsenault 1989) that active galactic nuclei are somewhat more likely to occur in galaxies having bars, than in those without.

Streamlines and periodic orbits

Because the velocity dispersion of the gas clouds is so much lower than their orbital speeds, the influence of “pressure” (collisions) on the trajectories will generally be small. When pressure is completely negligible, the gas streamlines must coincide with the periodic orbits in the system. However, gas streamlines differ from stellar orbits in one crucial respect: they cannot cross, i.e. the gas must have a unique stream velocity at each point in the flow. This very obvious fact implies that when periodic orbits cannot be neatly nested, pressure or viscous forces must always intervene to prevent gas streamlines from crossing.

Even when the perturbing potential is a weak, rotating oval distortion and orbits can be computed by linear theory, periodic orbits are destined to intersect at resonances. Not only do the the eccentricities of the orbits increase as exact resonance is approached, but the major axes switch orientation across all three principal resonances, making the crossing of orbits from opposite sides of a resonance inevitable. Sanders and Huntley (1976), using the beam scheme, showed that the gas response between the inner and outer Lindblad resonances takes the form of a regular two-arm spiral pattern in the density distribution. They argued that the orientation of the streamlines slews gradually over a wide radial range and the locus of the density maximum marks the regions where “orbit crowding” is greatest. Each spiral arm winds through, at most, 90° per resonance crossed. As Sanders and Huntley’s first model had a power-law rotation curve, only one ILR was present. In models having two ILRs, the orientation must change again through 90° at the inner ILR; note however, that we should expect a leading spiral arc at this resonance, because the dynamical properties of the orbits inside the inner ILR revert to those between the outer ILR and co-rotation. In a subsequent paper, Huntley et al. (1978) showed that the result in their case is a density response which leads the major axis of the potential by a maximum of about 45°. Where a weak bar potential rotates fast enough for no ILRs to be present, orbit crossings might be avoidable everywhere inside co-rotation. The flow may then remain aligned with the bar all the way from the centre to co-rotation (e.g. Schwarz 1981), changing abruptly at co-rotation to trailing spiral arcs extending to the OLR. It should be noted that all the results mentioned in this sub-section were obtained from a mild oval distortion of the potential having a large radial extent.

Strong bars

Streamlines still try to follow periodic orbits even in strongly non-axisymmetric potentials, though it becomes increasingly difficult to find circumstances in which the orbits can remain nested; not only can adjacent orbits cross, but a periodic orbit can also cross itself. Because this greatly complicates the relationship between periodic orbits and streamlines, we find the alternative picture described by Prendergast (1983), paraphrased here, easier to grasp.

As there is a formal analogy between compressible gas dynamics and shallow-water theory (e.g. Landau and Lifshitz 1987), we can think of the gas flow within a bar as a layer of shallow water circulating in a rotating non-axisymmetric vessel having the shape of the effective potential; for a rapidly rotating bar, this has the shape of a non-circular volcano crater.

As the crater rim defines co-rotation, the water within the crater flows in the same sense as the bar. It flows along the sides of the crater, but has too much momentum to be deflected round the end and back along the far side by the comparatively weak potentialgradients. Instead it rushes on past the major axis of the potential and on up the sides of thevessel, finally turning back when the flow stalls. The hydraulic jump which must form where fresh material encounters the stalled flow is the analogue of a shock in gas dynamics. This analogy provides an intuitive explanation for the location of shocks on the leading edge of the bar, something which is not so easily understood from the discussion in terms of periodic orbits presented by van Albada and Sanders (1983). As their main conclusion is that the periodic orbits must loop back on themselves, a condition implicit in Prendergast’s description, the two arguments are equivalent.

Strong offset shocks of this type were first revealed in the fluid dynamical simulations by Sørensen et al. (1976) and have been reproduced many times (Roberts et al. 1979, Sanders and Tubbs 1980, Schempp 1982, Hunter et al. 1988, etc.)

Athanassoula also concludes that the range of possible pattern speeds which gives rise to straight shocks is such that the major axis Lagrange points should lie between 1.1 and 1.3 times the bar semi-major axis, for a Ferrers bar model. If this result proves to be more general, and if the straight dust lanes are indeed the loci of shocks, then it supplies the tightest available constraint on the pattern speeds of bars in galaxies.

An additional result from her study is that the shocks are offset along the bar only when the potential supports a moderately extensive x2 (perpendicular) family of orbits. If the mass distribution is insufficiently centrally concentrated, then the shocks lie close to the bar major axis.

Angular momentum changes

Whenever gas is distributed asymmetrically about the major axis of the potential, it will experience a net torque which causes a secular change in its angular momentum. Where the density maximum leads the bar, the gas will systematically lose angular momentum, and conversely a trailing offset will cause it to gain. (The angular momentum is removed from, or given up to, the stellar population creating the non-axisymmetric potential.)

This process was emphasized by Schwarz (1981), who found that “gas” particles between co-rotation and the OLR were swept out to the OLR in a remarkably short time. The sweptup material quickly formed a ring, which was slightly elongated either parallel to the bar if the initial gas distribution extended to radii beyond the OLR, or perpendicular to it if the distribution was not so extensive. He obtained this result in an isochrone background potential using a bar pattern speed sufficiently high to have no ILRs, but in his thesis Schwarz (1979) also reports inner ring formation in a different model having an ILR.

Schwarz finds that the high rate at which gas is swept up into rings depends only weakly upon his numerical parameters: the collision box size, coefficient of restitution, etc. Since the torque responsible for these radial flows is proportional to the density contrast in the arms, as well as the phase lag (or lead) and strength of the non-axisymmetric potential, any realistic density contrast in the spiral or bar must give a similar flow rate. Schwarz’s flow rates are probably too high, however, because the bar-like potential perturbation he used, which peaks at co-rotation, is unrealistically strong in the outer parts.

Simkin et al. (1980) proposed a causal link between the inflow of gas within co-rotation and the existence of an active nucleus, which they suggest occurs somewhat more frequently in barred galaxies. This suggestion has been endorsed by Noguchi (1988), Barnes and Hernquist (1991) and others. While gas inflow along the bar is expected to raise the gas density in the central few hundred parsecs, its angular momentum must be further reduced by many more orders of magnitude before the material could be used to fuel a central engine.

Comparison with observations

NGC 5383 is probably the most extensively studied and modelled barred galaxy. Sanders and Tubbs (1980) made a systematic attempt to model the gas flow pattern measured by Peterson et al. (1978) and Sancisi et al. (1979). By varying the bar mass, axis ratio, pattern speed and other parameters they were able to find a model which broadly succeeded in reproducing the qualitative features of the observed flow pattern, though discrepancies in detail remained. An altogether more comprehensive attempt to model this galaxy was made by Duval and Athanassoula (1983), who used the distribution of surface brightness to constrain the bar density distribution and added more high resolution optical observations to map the flow pattern within the bar in more detail. They ran simulations to determine the flow pattern when co-rotation was close to the bar end and experimented mainly with a range of mass-tolight ratios for the bar; again their best model resembled the observed flow pattern within the bar, though still not impressively so. It is possible that their low resolution beam scheme code precluded a better fit.

Pence and Blackman (1984b) found that the velocity field of NGC 7496 closely resembled that of NGC 5383.

Following this initial success, a number of attempts have been made to model other galaxies, notably by the Florida group. One galaxy in their sample, NGC 1073 (England et al. 1990), is too nearly face on for the kinematic data to constrain a model. In both the other two, NGC 3992 (Hunter et al. 1988) and NGC 1300 (England 1989), they encountered considerable difficulties in modelling the outer spiral. When a pure bar model failed to produce a sufficiently strong density contrast in the outer spiral arms, they added a global oval distortion, but that seemed to produce too open a spiral pattern. It seems likely, therefore, that the spiral arms in these galaxies do not result from forcing by the bar.

NGC 1365 has also been observed extensively, though no good model for the whole galaxy has yet been published. Teuben et al. (1986) find quite convincing evidence for gas streamlines oriented perpendicularly to the bar near the very centre. They identify the location where this is observed with the x2 family of periodic orbits, giving them a rough indication of the pattern speed; the value obtained in this manner places co-rotation close to the end of the bar – a reassuring circumstance, which is also corroborated by the presence of the offset dust lanes along the strong bar. However, Ondrechen and van der Hulst (1989) note that the inward direction of the gas flow on the projected minor axis provides an unambiguous indication that the spiral arms at this point are still inside co-rotation. These two conclusions can be reconciled only by accepting that the bar and spirals are two separate patterns, with the bar rotating much faster than the spirals. Separate pattern speeds for the bar and spiral may occur in many galaxies. Co-rotation for the spiral pattern in NGC 1097 appears to lie beyond the bar (Ondrechen et al. 1989); Chevalier and Furenlid (1978) had difficulty in assigning a pattern speed for NGC 7723; the dust lanes in NGC 1365 and NGC 1300 (Sandage 1961) cross the spiral (another, much weaker indication of co-rotation) well beyond the end of the bar. The co-rotation resonance for the two tightly wrapped spiral arms which make up the pseudo inner ring of NGC 6300 appears to lie inside the point where they cross minor axis (Buta 1987), yet this is close to the bar end; this may be an example where the spirals rotate more rapidly than the bar – the misalignment between the spirals and the bar also supports the idea of separate patterns.

The conclusions from all these studies are:

i. Most radio observations need to be supplemented by high resolution optical data before modelling of the observed flow pattern provides useful constraints on the properties of the bar.

ii. The gas flow within the bar can be modelled fairly successfully when the bar pattern speed is about that required to place co-rotation just beyond the end of the bar.

iii. Shocks along the bar also develop under the same conditions for the bar pattern speed, but are offset only if the x2 family is present.

iv. The outer spiral arms usually cannot be modelled without assuming some additional non axis-ymmetric component to the potential.

v. Not one of the well studied cases provides evidence that the spiral patterns are driven by the bar, while there is frequently a suggestion that the spiral arms have a lower pattern speed than does the bar.

Origin of Bars

The first N-body simulations of collisionless stellar discs (Miller and Prendergast 1968, Hockney and Hohl 1969) revealed that it is easy to construct a rotationally supported stellar disc which is globally unstable and forms a large-amplitude bar on a dynamical timescale. While this instability offers a natural explanation for the existence of bars in some galaxies, it has proved surprisingly difficult to construct stable models for unbarred galaxies. The problem of the origin of bars in galaxies was therefore quickly superseded by that of how a fraction of disc galaxies could have avoided such an instability.

Though this last question is of only peripheral interest to a review of barred galaxies, it is hard to overstate its importance for disc galaxy dynamics. The bar instability has therefore been repeatedly re-examined from a number of directions, which have all tended to confirm that rotationally supported discs suffer from vigorous, global, bi-symmetric instabilities. It would take us too far from the subject of this review to discuss the bar instability in great depth, and we confine ourselves to a description of two aspects: the formulation of a global stability analysis for discs as an eigenvalue problem and the mechanism for the instability.

Bar-forming modes

The dominant modes found in many of these analyses have very high growth rates and an open bi-symmetric spiral form. In several cases the equilibrium model has also been studied in high quality N-body simulations which reveal a dominant mode with a shape and eigen-frequency in close agreement with the analytic prediction (Zang and Hohl 1978, Sellwood 1983, Sellwood and Athanassoula 1986). The possibility that N-body simulations somehow exaggerate the saturation amplitude, and hence the significance of the instability, was eliminated by Inagaki et al. (1984), who compared an N-body simulation with a direct integration of the collisionless Boltzmann equation for the same problem. The bars which form appear to be robust structures that survive for as long as the simulations are continued.

It seems likely that bars in real galaxies were created in this way, since many of their observed properties are similar to those of bars in the simulations (e.g. Sparke and Sellwood 1987), yet the majority of galaxies do not possess a strong bar. To account for this, we must understand how most galaxies could have either avoided this instability or subsequently regained axial symmetry. We cannot claim to have a completely satisfactory theory for the formation of bars in some galaxies unless we can also account for the fraction of galaxies that contain strong bars.

Properties of the resulting bars

This type of behaviour is typical of almost every two-dimensional simulation for which the underlying model is unstable to a global bi-symmetric distortion. As the instability runs its course, the transient features in the surrounding disc fade quickly and the only non-axisymmetric feature to survive is the steadily tumbling bar.

Many authors (e.g. Sellwood 1981, Combes and Sanders 1981, etc.) have observed that the bar ends at, or usually just inside, co-rotation (or more correctly the Lagrange point).

Thus, as the linear global mode saturates, its spiral shape straightens within co-rotation while the trailing arms outside co-rotation become more tightly wrapped and fade. Also the pattern speed of the bar immediately as it forms is generally close to that of the original global mode, in most cases slightly lower but sometimes higher. The figure rotation rate may quickly start to change as the model evolves further. The rule that the initial bar nearly fills its co-rotation circle appears to be widely held and no counter examples have been claimed. Sellwood (1981) also found that the bar length appeared to be related to the shape of the rotation curve, but his models were all of a particular type and different results are obtained from other models (Efstathiou et al. 1982, Sellwood 1989). The axis ratio of the bar largely depends upon the degree of random motion in the original disc: generally speaking, the cooler the initial disc the narrower the resulting bar (Athanassoula and Sellwood 1986).17 The most extreme axial ratios to survive for at least a few tumbling periods are in the range 4 – 5:1.

Sparke and Sellwood (1987) give a comprehensive description of one of these N-body bars. They emphasize that the bar is much more nearly rectangular than elliptical and matches the observed profiles remarkably well. The stars within the bars in these simulations exhibit a systematic streaming pattern, which again bears some resemblance to the observed velocity field.

Mechanism for the mode

The impressive agreement between the results from global analysis and the behaviour in N-body simulations leaves little room for doubt that rotationally supported, self-gravitating discs have a strong desire to form a bar. Yet these results do not explain why this instability should be so insistent, nor do they reveal the mechanism for the mode or give any indication as to how it could be controlled. Ideas to answer these questions have emerged from local theory, however.

Toomre (1981) proposed that the bar-forming mode was driven by positive feedback to an amplifier. The inner part of a galaxy acts as a resonant cavity which may be understood as follows:

(i) the group velocity of trailing spiral waves is inwards while that of leading waves is outwards,

(ii) trailing spiral waves that can reach the centre are reflected as leading waves,

(iii) a second reflection occurs at co-rotation where a leading wave rebounds as an amplified trailing disturbance. Standing waves can occur at only those frequencies for which the phase of the wave closes, which implies a discrete spectrum. Super-reflection off the co-rotation resonance causes the standing wave to grow, however, making the mode unstable. The super-reflection, which Toomre (1981) aptly named “swing amplification”, was first discussed by Goldreich and Lynden-Bell (1965) and by Julian and Toomre (1966). It has been further developed by Drury (1980) and invoked by Bertin (1983) and Lin and Bertin (1985). Wave action at co-rotation is conserved through a third, transmitted wave which carries away angular momentum to the outer galaxy.

An alternative theory of bar formation

While the focus is that bars are formed quickly as a result of some large-scale collective oscillation of the disc stars, we should note that an entirely different viewpoint was proposed by Lynden-Bell (1979).

In a brief, but elegant paper, Lynden-Bell suggested that bars may grow slowly through the gradual alignment of eccentric orbits. He pointed out that a bar-like perturbing potential would exert a torque on an elongated orbit close to inner Lindblad resonance. He then showed that the torque would cause the major axis of the orbit to oscillate about an axis orthogonal to the density perturbation in outer parts of galaxies, but near the centre, in regions where double ILRs are possible, the orbit would tend to align with the perturbation.

In the aligning region, orbits having very similar frequencies would tend to group together at first, creating a bi-symmetric density perturbation which would then exert sufficiently strong tangential forces to trap orbits whose frequencies differ somewhat from that of the bar. A large perturbation can gradually be assembled in this manner over many dynamical times. Significantly, the pattern speed of the bar would also decline as the orbits become more eccentric, enabling yet lower frequency orbits to be caught by the potential. The excess angular momentum which needs to be shed from the barring region would be carried away by spiral arms. Such a bar would not extend as far as its co-rotation point.

Although Lynden-Bell was originally motivated to understand the weak bar-like features which developed in the bar-stable simulations of James and Sellwood (1978), he proposed that the mechanism describes the general evolution of most galaxies. Given the ease with which bars are formed by fast collective instabilities, it is unclear whether this mechanism plays any role in the formation of the strong bars we see in galaxies today. However, the alternative view of interacting orbits presented in his paper does provide considerable insight into the structure of a bar.

This mechanism gives rise to the linear bi-symmetric modes of a radially hot, non-rotating disc calculated by Polyachenko (1989). In the appendix of their paper, Athanssoula and Sellwood (1986) described a few vigorous bi-symmetric modes in their simulations of hot disc models which, however, saturated at a very low amplitude. If these modes were indeed of the type suggested by Polyachenko. based on Lynden-Bell’s mechanism, the instability is probably unrelated to the formation of strong bars.

Conclusions

It should be clear that we are still far from a complete understanding of the basic structure of barred galaxies, but progress has been rapid in recent years – especially where orbit studies have been connected to observational or simulated results. There are a few aspects where the observed facts seem to fit reasonably well with theoretical ideas.

On the bright side, it seems very likely that strong bars are formed by the dynamical instability. Bars formed in the N-body experiments end near co-rotation and have mass distributions and kinematic properties which seem to correspond with those observed, though more detailed comparisons would be desirable. Moreover, such bar models appear to have the right strength, and to rotate sufficiently rapidly to shock gas in places resembling the dust lane patterns in some barred galaxies. The flow patterns are understood in terms of orbit theory and fit the observed kinematics for entirely plausible model parameters. We also think we understand the origins of rings sufficiently well to be able to interpret them as signatures of major resonances with the bar pattern.

These successes add up to a compelling, but indirect, case for a bar pattern speed in galaxies which places the major axis Lagrange point just beyond the bar’s end. Such a value is consistent with the more direct observational estimates of pattern speed which unfortunately are subject to large uncertainties.

Current observational work is beginning to provide more quantitative data on the light distributions and kinematics of barred galaxies. In particular, the kind of comparison with theoretical models made by Kent and Glaudell (1989) for NGC 936 should be extended to several other galaxies in case the structure of that galaxy is special in any way. Moreover, the theory of ring formation is in dire need of more detailed observational comparisons; we need good kinematic maps of many more such galaxies to confirm that the rings do indeed lie at the resonances for the bar.

Yet our understanding of the dynamical structure of bars is still far from complete. The vast literature on orbits in bar-like potentials has led to a few major conclusions relevant to the structure of barred galaxies: the most important is that the majority of stars within the bar probably follow eccentric orbits which are trapped, or semi-trapped, about the main family of orbits aligned with the bar (the x1 family). Apart from this, knowledge of the main orbit families has improved our understanding of gas flows and indicates that it would be hopeless to try to construct a self-consistent bar having a mean rotational streaming in a sense counter to the pattern rotation. It is progress, of a kind, to learn that a simple generalization of the two-dimensional orbital structure does not provide a complete description of three-dimensional bars. Only one set of approximately self-consistent two-dimensional numerical solutions has been found (Pfenniger 1984b) and the prospects for analytic models in the near future are bleak. Next to this, our fragmentary understanding of the evolution of barred galaxies, and almost total blanks on the origins of lenses and the pronounced asymmetries in some galaxies, seem of secondary concern.