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# LEADERSHIP LABORATORY M S 0

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Reports on Progress in Physics 56 1993 173255 Dynamics of Barred Galaxies J A Sellwood1gt2 and A Wilkinson3 1Space Telescope Science Institute 3700 San Martin Drive Baltimore MD 21218 USA 2Department of Physics and Astronomy Rutgers University PO Box 849 Piscataway NJ 088550849 USA 3Department of Astronomy The University Manchester M13 9PL England Abstract 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 centre 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 sum marize 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 rst results suggest that bars should be thicker in the third dimension than the disc from which they formed Gas ow 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 Submitted July 1992 accepted November 1992 appeared February 1993 ii Sellwood and Wilkinson Table of Contents 1 2 Cf 03 I Introduction Observed properties of bars 21 Components of the light distribution 22 Fraction of total luminosity in the bar The light distribution within the bar Tri axial bulges andor nuclear bars 25 Kinematic properties of bars Bar angular velocity or pattern speed Velocity dispersions Stellar dynamics of galaxies 31 Relaxation time 32 Dynamical equations 33 Analytic distribution functions 34 Near integrable systems 35 Linear programming 36 N body techniques Two dimensional bar models 41 Thin disc approximation 42 Linear theory 43 Strong bars 44 Periodic orbits 45 Periodic orbit families 46 Non periodic orbits 47 Onset of chaos 48 Actions 49 Self consistency Three dimensional bar models 51 Vertical resonances 52 Per 00 3 Barred galam39es Asymmetries 81 Observed properties 82 Models Origin of bars 91 Global analysis 92 Bar forming modes 93 Properties of the resulting bars 94 Mechanism for the mode 95 Controlling the bar instability 96 Meta stability and tidal triggering 97 An alternative theory of bar formation Evolution of the bar 101 Buckling instability 102 Peanut growth 103 Continuing interactions with the disc 104 lnterations with spheroidal components 105 Destruction of bars Conclusions Acknowldegments References Barred galam39es 1 1 Introduction Galaxies are beautiful objects7 and the graceful symmetry of many barred galaxies is particu larly striking One of the best examples close enough for us to examine in detail is NGC 1365 in the Fornax cluster Figure 1 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 objects7 as can be seen from the selection assembled in Figure 2 The bar component ranges from a minor non axisymmetric perturbation to a major feature in the light distribution Other properties7 such as the size of the bar relative to the host galaxy7 the degree of overall symmetry7 the existence of rings and the numbers and position relative to the bar of spiral arms in the outer disc7 the gas and dust content7 etc7 vary considerably from galaxy to galaxy Bars can be found in all types of disc galaxies7 from the earliest to the latest stages of the Hubble sequences Because there is a continuum of apparent bar strengths from very weak oval distortions to major features7 it becomes more a matter of taste to decide what strength of bar in a galaxy is suf cient to merit a barred classi cation Figure 3 shows the fraction of each spiral type judged to contain bars7 compiled from A Revised Shapley Ames Catalogue Sandage and Tammann 19817 hereafter RSA7 The Second Reference Catalogue de Vaucouleurs et al 19767 hereafter RC2 and the Uppsala General Catalogue Nilson 19737 hereafter UGC Apart from the very latest types7 there is rough agreement over the fractions containing strong bars shaded when combined over all stages7 the SB family constitutes between 25 and 35 of all disc galaxies However7 there is considerably less agreement over intermediate cases7 the SAB family the morphological classi cations assigned in the RC2 indicate a combined fraction for the SAB family as high as 264 7 substantially higher than in either of the other two catalogues Notwithstanding these variations7 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 19837 NGC 1068 Scoville et al 19887 Thronson et al 1989 and NGC 309 Block and Wainscoat 1991 much shorter or weaker bar like features have shown up in many others 24 Many more such cases could yet be discovered7 as lR cameras are in their infancy if so7 the fraction of barred galaxies could turn out to be much higher than indicated by Figure 3 Elliptical galaxies are sometimes described as stellar bars7 which is apt for certain purposes However7 we exclude such galaxies from this review because7 unlike barred galaxies7 they appear to be single component systems with no surrounding disc or distinct central bulge Moreover7 the stellar distribution is not thought to be as attened7 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 3 Galaxies are classi ed by their morphological appearance in a system based upon that devised by Hubble 1926 the relative luminosity of the central bulge and openness of the spirals are used to de ne a sequence S07 Sa7 Sb7 Sc7 Sd for both barred and unbarred galaxies Early type galaxies S07 Sa have bright bulges and weak7 tightly wrapped spirals late types Sc7 Sd have little or no bulge and open7 ragged spirals Sellwood and Wilkinson Figure 1 NGC 13657 a magni cent barred spiral in Fornax The galaxy has a very bright amorphous central bulge7 a strong bar joined to a nearly symmetrical pair of spiral arms Pronounced brown dust lanes can be seen in the spiral arms and on the leading edge of the bar assuming the arms to be trailing Sites of star formation are tinged in pink While the youngest stars are blue and old stars are yellow As in most galaxies7 the spiral arms are superposed on a much lower surface brightness disc7 which is too faint to be distinguishable from the sky in this photograph Photograph courtesy of David Malin7 Anglo Australian Observatory Barred galaxies Figure 2 Eight further examples chosen to illustrate the variety of galaxies classi ed as barred They are a NGC 936 b NGC 1433 c NGC 2523 d NGC 1398 e NGC 2217 f NGC 5236 M83 g NGC 5383 h NGC 1313 Seven photographs are courtesy of Allan Sandage and the Carnegie Institution The eighth NGC 1398 d which was supplied by David Malin and has been processed to bring out srnall scale structure Sellwood and Wilkinson Figure 2 Continued Barred galaxies 5 PCZ UGC Percent b0 rred SEOSBOCI SBGSBUD SBbSBbc SBCSBCd SBdSBmle Figure 3 The fractions of barred SB shaded and intermediate SAB unshaded types at di erent stages along the Hubble sequence of spiral galaxies identi ed in three independent morphological classi cations The statistics from the RSA are based upon 987 objects only the rst classi cation was used here The RC2 sample contains 1339 objects after excluding peculiar uncertain and spindle types edgeion objects those with axis ratios gt 3 and those with diameters less than 1 arcmin 7 e s largescale plate material was available The UGC sample contains 4169 galaxies when selection criteria similar to those for the RC2 were applied The statistics are insensitive to the selection criteria provided they remain within sensible ranges to be able to calculate the gravitational potential This information combined with the rotation rate of the bar would enab e us to calculate the motion of stars an gas We could claim we understood the dynamical equilibrium if we were able to propose a selficonsistent 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 rst 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 discilike we might naively hope that there is a plane of symmetry but because barred galaxies are intrinsically none circular determination of the inclination angle is more dif cult than in their unbarred counterparts ii Worse still it is unlikely that a single plane of symmetry exists in many galaxies Those seen edgeion frequently exhibit warps mainly in the outer parts yet this is precisely where the problem of point 2 is best avoided 6 S6llw00d and Wilkinson 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 7 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 lt 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 69 Disney 6t al 1989 Valentijn 1990 Huizinga and van Albada 1992 6136 Our snapshot77 view of each galaxy prevents our making direct measurements of the rota tion 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 S Many features 69 dust lanes and rings result from dissipative processes in the gaseous component Unfortunately we lack a good theoretical description for the large scale be haviour 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 di ferent distribution from the light There is some evidence that the luminous matter dominates the dynamics of the inner parts of unbarred galaxies 69 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 25 amp 6 that the more prominent bars are associated with large departures from simple circular motion in the disc plane indicating that the bar contains a signi cant fraction of the mass in the inner galaxy 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 5 a 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 andor rings 69 Kormendy 1979 Most of these components are evident from Figures 1 X6 2 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 pro le Although the working hypothesis of separate components seems justi ed by the fact that the individual features are readily distinguished by the eye it is much more dif cult to separate the components quantitatively 21 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 rst part of the review 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 suf cient 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 Barred galtwies 7 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 com pleted some 50 orbits around the centre 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 re ect initial transients It is most appropriate7 therefore7 to consider the slow evolution of a nearly equilibrium model Ac cordingly 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 2 Observed properties of bars We begin by summarizing the observational information relating to the dynamical structure of the stellar bar7 and leave data on gas to 6 and on rings to 7 where we discuss these other phenomena Unfortunately we cannot make direct measurements of even the most basic properties7 and are forced to make indirect inferences from the observables Our requirements fall into three general areas i The distribution of mass7 in order to determine the gravitational potential This has to be deduced from measurements of a the light distributions and b the velocity eld ii The bar rotation velocity7 or pattern speed7 which can be inferred only with considerable uncertainty from the velocity eld or7 still less directly7 through modelling the gas ow pattern iii The full three dimensional velocity dispersion which7 jointly with orbital streaming7 de termines the stellar dynamical equilibrium This can be constructed from the projected velocity dispersion data only with the aid of a mass model 2 Components of the light distribution Deprojection of the light distribution of barred galaxies is more dif cult than for nearly axisym metric 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 circles7 but it should be borne in mind that even far out the shape could be intrinsically elliptical7 especially if an outer ring is present 77 or the plane is warped The major axis of the bar is always less than the diameter of the host galaxy ln general7 bars in late type systems are shorter relative to the total galaxy size DbarD25 02 to 03 than those in early type galaxies DbaID25 03 to 067 where D25 is the diameter at which the surface brightness of the galaxy falls below 25 mag arcsec Z eg Athanassoula and Martinet 19807 Elmegreen and Elmegreen 19857 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 N 234 times the bulge diameter7 while Baumgart and Peterson 1986 estimate 26 i 07 22 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 8 S6llw00d and Wilkinson 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 69 D25 Decompositions were originally performed by attempting to t the components with ide alized models prolate bars exponential discs 6t6 69 Crane 1975 Okamura 1978 Duval and Athanassoula 1983 Duval and Monnet 1985 Blackman 1983 used breaks or changes of slope in the photometric pro le to de ne the boundaries between the different components Others have tried Fourier analysis of the light distribution 69 Elmegreen and Elmegreen 1985 Buta 1986b 1987 Ohta 6t 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 6t al 1990 give values in the range 25 to 50 in the bar region77 to 10 to 30 out to D25 for their sample of early type galaxies 23 Th6 li9ht distribution within th6 bar Generally the shapes strengths and lengths of bars seem to vary systematically from early to late type systems 23 Early typ6 9alami6s Many bars in galaxies of types SBO and SBa have a pronounced rectangular shape seen in projection Ohta 6t al 1990 Athanassoula 6t al 1990 with axial ratios between 03 to 01 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 69 Elmegreen and Elmegreen 1985 Kent and Glaudell 1989 There is no difference between the leading and trailing sides Ohta 6t 11 The surface brightness contrast between bars and the axisymmetric component can range from 25 to 55 Ohta 6t 11 Bars are considerably brighter on the major axis and usually fainter on the minor axis than the inwardly extrapolated disc pro le Azimuthally averaged however the radial pro les are no more varied than those of ordinary spirals Wozniak and Pierce 1991 t an exponential to the disc pro le outside the bar but ring features which are much more common in barred galaxies can make this a poor approximation 69 Buta 1986b Lat6 typ6 9alacci6s The light distribution in late type galaxies is generally less smooth owing to greater dust obscuration and more intense knots of young stars The dif culties created by both these problems are lessened by making the observations at wavelengths as far into the infra red as possible 69 Adamson 6t al 1987 Bars in galaxies of types SBbc to SBm are generally more elliptical 69 Duval and Monnet 1985 shorter and weaker than their earlier counterparts They also begin to show quite pronounced asymmetries 69 one end may appear squarer than the other see 8 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 Barr6d 9alai6s 9 Elmegreen and Elmegreen 19857 Baumgart and Peterson 1986 Across the bar7 the light pro le is close to Gaussian 69 Blackman 1983 Li9ht distribution normal to th6 galactic plan6 Although we cannot see them7 we expect bars to be as common in edge on systems as in all disc galaxies It has therefore been argued that since we do not see a large fraction of unusually thick discs7 bars must be as thin as the rest of the disc A range of values has been suggested for the bar thickness Kormendy 1982 suggests typical axis ratios of 1210 and extremes of perhaps 1315 or 134 69 Burstein 19797 Tsikoudi 19807 1z10 Wakamatsu and Hamabe 1984 However7 this argument could be fallacious The bulges at the centres of a signi cant fraction of nearly edge on galaxies do not have a simple spheroidal or ellipsoidal shape7 but are squared off7 boxy or even indented peanut shaped 69 Sandage 1961 plate 77 Jarvis 19867 Shaw 19877 de Souza and dos Anjos 1987 It has been suggested Combes and Sanders 19817 Raha 6t al 1991 that such bulges may be the signature of a bar seen edge on7 and if so bars are much thicker than discs We discuss this point further in 102 Bottema 1990 has measured the velocity dispersion pro le of a bar which we see nearly face on He nds that the dispersion7 which is dominated by the vertical component7 remains approximately constant at N 55 km s 1 from just outside the bulge to the end of the bar As the light pro le declines strongly along this late type bar7 we might expect the surface mass density to do so also Data of this kind should be examined more closely to see whether they imply a aring bar 24 Tii accial bul96s and07quot nucl6aiquot bars The position angle of the major axes of isophotes in the inner regions of many barred galaxies twists away from the bar major axis near the centre The twist angles are substantial and may7 when deprojected7 be consistent with a central elongation perpendicular to the bar Such features have been seen mainly in the optical 69 de Vaucouleurs 19747 Sandage and Brucato 19797 Buta 1986b7 Jarvis 6t al 1988 but also in the infrared Baumgart and Peterson 19867 Pierce 19867 Shaw 6t al 19927 in molecular gas Ball 6t al 19857 CanZian 6t al 19887 lshizuki 6t al 19907 Kenney 1991 and even radio continuum Hummel 6t al 1987a amp b It has frequently been suggested that such features indicate a tri axial7 rather than an axially symmetric7 bulge 69 de Vaucouleurs 19747 Kormendy 19797 19827 Gerhard and Vietri 19867 and many others Others have claimed it indicates a small nuclear bar7 in both barred and unbarred galaxies As there are theoretical reasons to expect twisted isophotes near the bar centre see 451 amp 64 it is of some importance to determine whether the inner feature is perpendicular to the bar major axis when deprojected to face on While this dif cult question is not yet settled7 there is evidence that the features are frequently almost perpendicular7 but Gerhard and Louis 1988 argue that this cannot be so in all cases 25 Kin6mati6 pr0p6i39ti6s of bars Material in galaxies lacking a bar7 or other strong non axisymmetric feature7 seems to follow nearly circular orbits7 and the velocity eld is simply that of rotational motion seen in projec tion Barred galaxies7 on the other hand7 are more complicated for two reasons rstly7 the velocities7 especially in the barred region7 manifest strong non circular streaming motions7 and secondly7 the viewing geometry is much more dif cult to determine Both the inclination angle 10 Sellwood and Wilkinson to the plane of the sky and the position angle of the projection axis 7 the line of nodes 7 are more dif cult to determine when the light distribution is intrinsically non axisymmetric and the motions depart systematically from circular Many early observations consisted of a slit spectrum at a single position angle along the bar or projected major axis Such data yields very incomplete information several slit positions7 or ideally a full two dimensional map7 are required to determine the velocity eld 25 Velocity elds Kormendy 1983 presented the rst clear evidence for non circular stellar streaming motions in a barred galaxy The non circular motions in the SB0 galaxy NGO 936 Figure 2 are about 20 of the circular streaming velocity and are consistent with orbits being elongated along the bar7 with the circulation in the forward sense along the bar major axis assuming the weak spiral arms trail More recent data for the same galaxy Kent 19877 Kent and Glaudell 1989 have con rmed this interpretation Other examples are NGO 6684 which shows deviations of 100 kmsec from circular motion Bettoni et al 19887 NGO 15437 15747 4477 for which velocities on the minor axis of 100 km s 1 have been detected Jarvis et al 1988 and NGO 4596 Kent 1990 Only galaxies for which the bar is inclined to both the projected major and minor axes show non circular motions clearly Pence and Blackman 1984b7 Long 1992 Bettoni and Galletta 1988 argued that the picture is more complicated than simple stream ing in the same sense as the direction of bar rotation because the apparent streaming velocity does not always rise monotonically when measured along the bar major axis and may even dip back to near zero at a point about one third of the distance from the centre to the end of the bar They suggested that this behaviour might possibly be due to retrograde stellar streaming within the bar However7 Sparke showed that it can quite naturally be explained as being due to the nite width of the slit7 and a possible very small misalignment with the bar major axis7 allowing light from stars streaming along the bar to in uence the measurement Her argument is reported by Bettoni 1989 These measurements establish that the bar distorts the axisymmetric potential of the disc suf ciently strongly to force the stars to stream on highly elliptical orbits inside the bar Addi tional evidence which helps to determine the strength of the non axisymmetric potential comes from observations of the gas kinematics7 though these usually need to be interpreted using a model see 6 252 Azimuthally averaged velocity pro les Since the potential of a barred galaxy is strongly non axisymmetric7 it is even more dif cult than for a nearly axisymmetric galaxy to determine the distribution of mass Although an approximation to a rotation curve77 found by crudely averaging tangential streaming velocities can be signi cantly in error Long 19927 this practice is widespread The error is unlikely to affect the comment Elmegreen and Elmegreen 1985 that many early type galaxies have bars which extend beyond the rising part of the rotation curve7 while late type bars end before the rotation curve attens off However7 the positions of resonances within the bar see 42 deduced from an axially symmetrized rotation curve and an estimate of the bar pattern speed7 are highly uncertain Not only could the axisymmetric rotation curve be wrong7 but a resonance predicted by linear theory for a weak perturbation could be absent in a strong bar Orbit integrations are the only reliable way to identify resonances in a strong bar 45 Barred galaries 11 26 Bar angular velocity or pattern speed The rate of rotation of the bar is one of the most important parameters determining its dynam ical structure but unfortunately it is very dif cult to measure directly Tremaine and Weinberg 1984b have shown how the continuity equation could in principle be used to determine this quantity from observables alone The method requires high signal to noise long slit spectra and a photometric light pro le both taken along cuts offset from the nucleus and parallel to the major axis In practice the pattern speed inferred is sensitive to centering and alignment errors warps and the presence of any non bisymmetric perturbation and is therefore uncertain Using this technique Kent 1987 estimated 9 sini 54 i 19 km s larcsec 1 104i37 km s lkpc l for NGC 936 Combining his value with Kormendy7s rotation curve Kent7s measurement suggests that co rotation lies within the bar although a radius just beyond the end of the bar is within the error4 Application of this method to galaxies other than SBOs is even more problematic because of the presence of dust patches and bright knots Tremaine and Weinberg tried to apply it to the H1 observations of NGC 5383 Sancisi et al 1979 but were unable to obtain a conclusive result The method failed because of noise an extended Hl envelope and because any single easily observed species probably does not obey the equation of continuity eg atomic to molecular transitions star formation stellar mass loss etc The very limited success of this method is disappointing nevertheless Kent7s measurement does at least furnish independent evidence that the bar in NGC 936 rotates rapidly We are therefore forced to rely upon less direct alternative techniques The most reliable method described in 6 attempts to match the observed gas ow pattern to a set of hydrodynamical models in which the bar pattern speed is varied Some authors also attempt to associate indi vidual morphological and kinematic features with resonances for the pattern we note examples in 67 amp 7 27 Velocity dispersions Velocity dispersion pro les have also been determined for some galaxies though frequently only along the bright bar major axis The central values of the velocity dispersion are found to be similar to that of the lens 7 7 typically about 150 to 200 km s 1 eg Kormendy 1982 for the SB0 galaxies NGC 936 and NGC 3945 The ratio of the maximum rotation velocity to the central velocity dispersion Vmaxoo gives a measure of the relative importance of streaming and random motion Values are typically between 04 and 05 implying that there are signi cant contributions from both streaming motions along the bar and random motions More generally the bar velocity dispersion pro les can vary from at to sharply falling with increasing radius eg Jarvis et al 1988 3 Stellar dynamics of galaxies In this section we summarize the basic equations of stellar dynamics and some of the techniques used to solve them The following two sections discuss the structure of steady bars in two and 4 Notwithstanding his measurement Kent and Glaudell 1989 preferred a pattern speed 64 i 15 km s lkpc 1 in their dynamical study of this galaxy 473 in order to place corotation just outside the end of the bar 12 Sellwood and Wilkinson three dimensions respectively An excellent introductory text has been provided by Binney and Tremaine 1987 31 Relaccatizm time Since the density of stars throughout the main body of a galaxy is very low7 the orbit of an individual star is governed by the large scale gravitational eld 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 relamatizm time is the time taken for these cumulative random de ections to change the velocity components along the orbit of a typical star by an amount equal to the stellar velocity dispersion Chandrasekhar7s 1941 formula yields a value of 1013 years7 or N 1000 times the age of the universe7 for star star encounters in the neighbourhood of the Sun Since the relaxation time varies inversely as the stellar density7 considerably shorter timescales apply in the centres of galaxies7 where the star density is higher by several orders of magnitude Chandrasekhar7s formula has not required revision7 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 7 gas is accumulated into giant molecular cloud complexes7 ranging in mass up to 105 MG or even 106 MG and some bound star clusters are known to contain almost as much mass in stars Although these objects are much more diffuse than point masses7 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 material7 known as a polarization cloud7 can easily exceed the mass of the perturber when the stellar motions are highly ordered 7 as in a rotationally supported disc Julian and Toomre 1966 Nevertheless7 the relaxation time in the objects we consider here is never shorter than the orbital period7 and is usually considerably longer 32 Dynamical equations The long relaxation time implies that the distribution of stars in a galaxy approximates a collisionless uid and therefore obeys dynamical equations similar to those governing a Vlasov plasma The closest parallel is with a single species plasma eg Davidson et al 1991 To describe such a uid mathematically7 we introduce a distribution function7 F7 which is the density of particles in an element of phase space 7 a multi dimensional space with generally three spatial dimensions and three velocity dimensions The de nition of F 2 7 at is the mass within a volume element of phase space7 divided by the volume of that element7 and we would like to take this to the limit of in nitesimal volume As the number of stars in a galaxy is nite7 however7 F can be de ned only for volume elements large enough to contain many stars Since this can be very inconvenient mathematically7 we generally imagine that the number of stars is greatly in nitely increased7 while their individual masses are correspondingly reduced7 so that F can be meaningfully de ned for in nitesimal volume elements Because encounters are negligible7 the masses of individual stars do not enter into the equations The motion of a uid element in this phase space is governed by the collisionless Boltzmann or Vlasov equation 62vVF7Vlt1gt6F7 at E 7 039 1 Barred galacics 13 Since the left hand side is the convective derivative in this space equation 2 simply states that the phase space density is conserved at a point which moves with the ow The quantity ltIgt in 2 is the smooth gravitational potential of the galaxy which is related to the volume density p through Poisson7s equation V2lt1gtz t 47erz t 2 where G is Newton7s gravitational constant The volume density of stars at any point is the integral of the phase space density over all velocities pm t L Fxvtd339v 3 The right hand side of equation 2 is zero only for a collisionless uid Rapid spatial or temporal variations about the mean potential of whatever origin can scatter stars in phase space and lead to a non zero Fokker Planck term here cg Binney and Lacey 1988 We assume for the purposes of this review that no such term exists The eld of galactic dynamics is to a large extent concerned with solutions to this coupled set of integro di erential equations Only a few analytic equilibrium solutions are known in cases where a particularly simple symmetry cg spherical axial planar etc has been assumed These and in some cases the stability of the resulting models are discussed in the monograph by Fridman and Polyachenko 1984 33 Analytic distribution functions The classical approach to solving these equations for a particular mass distribution is rst to identify the integrals ic quantities that are conserved by a test particle pursuing any orbit in the potential In a steady non rotating potential the particle conserves its energy and if the potential has an axis of symmetry the component of angular momentum about that axis is conserved Jeans7s theorem tells us that the distribution function for an equilibrium model is a function of the integrals of the motion cg Lynden Bell 1962a The identi cation of integrals with simple physical quantities such as energy or angular momentum is possible only for very simple potentials and the approach rapidly becomes too hard for realistic rotating non axisymmetric potentials Vandervoort 1979 and Hietarinta 1987 discuss the possible forms an extra integral could take in a bar like potential One possibility had already been exploited by Freeman 1966 who proposed a family of two dimensional bar models in a rotating potential consisting of two uncoupled harmonic oscillators His highly idealized models have some interesting properties but all have retrograde streaming of the stars inside the bar which con icts with the observed situation 25 Vandervoort 1980 developed a set of stellar dynamical analogues to the Maclaurin and Ja cobi sequences of gaseous polytropes These are uniformly rotating non axisymmetric isotropic bodies for which the distribution function is a function of one integral only These models are not centrally concentrated enough for elliptical galaxies but this may not be such a drawback for strong bars in which the density is less centrally concentrated The most promising recent bar like models for which some headway has been and still is being made are the so called Stackel models which emerged from Hamilton Jacobi theory cg Lynden Bell 1962b Stackel 1890 identi ed a class of potentials in which the Hamilton Jacobi equation separates in confocal ellipsoidal coordinates and in which the density is strati ed on 14 Sellwood and Wilkinson concentric tri axial ellipsoids Their main advantage for theoretical work is that they have the most general form known to be always integrable see 34 This implies that once the distribution function is known all the observable properties of the model such as the streaming velocities and velocity dispersion elds are predictable The principal disadvantage for our subject is that almost all such models so far discussed are non rotating These models have been extensively explored by Eddington 1915 Kuz7min 1956 Lynden Bell 1962b and de Zeeuw 1985 and his co workers as models for galaxies The central dense part can have a surprisingly wide variety of shapes de Zeeuw et al 1986 rang ing from simple ellipsoids to more rectangular bodies with squared off ends Unfortunately the distribution functions required to construct self consistent models are not so easy to obtain Statler 1987 obtained a variety of approximate solutions using Schwarz schild7s method 35 for several three dimensional ellipsoidal models5 Hunter et al 1990 derived exact expressions for the distribution function for arbitrarily attened prolate models populated by thin tube orbits Their solutions can be tailored to allow maximal or minimal streaming around the symmetry axis Hunter and de Zeeuw 1992 have given solutions for the tri axial case again they found a wide range of solutions but were able to show that self consistent models require some members from all four of the main orbit families One integrable bar model noted by Vandervoort 1979 as a curiosity was later shown to be the rst known rotating Stackel model Contopoulos and Vandervoort 1992 The centrifugal term is exactly balanced by a term in this highly contrived potential which has the unphysical features of two singularities at the co ordinate foci and negative mass density at large distances The orbital structure of the model studied in depth by Contopoulos and Vandervoort is quite different from that found in more realistic bar models Very little is known about the stability of Stackel models though Merritt and his collabo rators have initiated a programme to explore this issue Merritt and Stiavelli 1990 found that lop sided instabilities developed in all oblate models with predominantly shell orbits Prolate models on the other hand were unstable to buckling modes only when highly elongated Mer ritt and Hernquist 1991 This numerical approach complements stability analyses which are possible only for extremely simple models eg Vandervoort 1991 34 Near integrable systems The basic assumption underlying the approach just outlined is that bars are integrable systems ie that all stars possess the number of integrals equal to the number of spatial dimensions This now seems most unlikely as we show in 4 amp 5 galactic bars furnish an excellent practical example of the kind of near integrable Hamiltonian system rst studied by Henon and Heiles 1964 also in an astronomical context The study of near integrable systems opens up the modern topic of non linear dynamics Unfortunately even a highly simpli ed introduction would represent too large a digression for this review and we refer the reader to a standard text eg Lichtenberg amp Lieberman 1983 The identi cation of bars as non integrable systems not only raises the theoretical problem of why they are not integrable but also a number of practical questions The most important are what is the fraction of chaotic orbits within the bar how much does their density dis tribution contribute to the bar potential and what are the long term consequences of their 5 Teuben 1987 provides solutions for the twodimensional analogue the elliptic disci Barred galam39es 15 presence These questions remain largely unanswered and it should be clear that they represent a formidable obstacle to progress in our understanding of bar dynamics The eld has therefore resorted to the less elegant numerical methods described in the next two subsections in order to make further progress 35 Linear programming Schwarzschild 1979 achieved a major breakthrough when he successfully constructed the rst numerical models of a self consistent non rotating tri axial stellar ellipsoid without requiring any knowledge of the nature or even the number of integrals supported by the potential He began by computing a large number of orbits in a potential arising from an assumed tri axial ellipsoidal density distribution and noted the time averaged density along each orbit in a lattice of cells spanning the volume of the model He then used linear programming techniques to nd the non negative weights to assign to each orbit in order that the total density in each cell added up to that in his initially assumed model As he was able to nd more than one solution Schwarzschild added a cost function77 to seek solutions with maximal streaming for example However the method yields only an approximate equilibrium model even when a ne lattice of cells is used and gives no indication of its stability Newton and Binney 1984 proposed a variation to this strategy based upon Lucy7s iterative deconvolution technique Its main advantages are that it is easy to program fast and the resulting distribution function is smoother than that yielded by linear programming Other iterative algorithms are based upon non negative least squares Pfenniger 1984b and maximum entropy Richstone 1987 36 N body techniques Because of the intractability of the full blown self consistency problem many workers in the eld have turned to N body simulations Not only does a quasi steady N body model represent a self consistent equilibrium but it also demonstrates that the model is not catastrophically unstable Such models have yielded many important results and have contributed inestimably to our understanding of barred galaxies We present these results at appropriate points in this review and con ne ourselves here to alerting the reader to some of the essential limitations of the N body techniques Simple restrictions of computer time and to a lesser extent memory limit the numbers of particles that can be used in the simulations lnevitably therefore the potential in simulations is less smooth than in the system being modelled Statistical uctuations in the particle density on all scales from the inter particle separation up to the size of the system are larger than they ought to be by the square root of the ratio of the numbers of particles in some of the most ef cient high quality simulations which employ perhaps a few times 105 particles the noise level remains about one thousand times higher than in a galaxy of stars6 The various techniques available to combat this problem have been reviewed by Sellwood 1987 but only one the quiet start77 method removes uctuations on the largest scales The technique involves choosing the initial coordinates of particles in some symmetric con guration and ltering out the components of the potential which arise from the imposed symmetry 6 Density uctuations of this magnitude may be present in the discs of gas rich galaxies howeveri 16 Sellwood and Wilkinson Sellwood 1983 Unfortunately the symmetric con gurations themselves are unstable and disintegrate quite rapidly the respite from noise is therefore only temporary The main effect of these irrepressible uctuations is to scatter particles away from the paths they would have pursued in a noise free potential The integrals which should be invariant along the orbit of each particle are no longer preserved There have been comparatively few studies of orbit quality in simulations but enough to raise substantial worries 69 van Albada 1986 Hernquist and Barnes 1990 The simulations are thought to be useful for longer than the scattering time of a typical orbit because the individual scattering events when averaged over the whole population of particles accumulate only statistically However the long term stability and any possible secular e fects cannot be studied by these techniques For obvious reasons of computational economy the particles in a large majority of bar unstable disc simulations have been con ned to a plane A few early attempts to include full three dimensional motion Hockney and Brownrigg 1974 Hohl 1978 Sellwood 1980 re vealed behaviour which di ered only slightly from models having equivalent resolution in two dimensions but the grids used in all these calculations were so coarse that the out of plane motions were virtually unresolved The results obtained by Combes and Sanders 1981 did reveal an important di ference which was not understood at the time As extra resolution in two dimensions seemed essential Sellwood 1981 full three dimensional simulations for this problem were considered unnecessary for several years However recent three dimensional models have rediscovered Combes and Sanders7 old result and show that the two dimensional models miss some of the essential physics 51 amp 101 4 Twodimensional bar models We begin our detailed discussion of the dynamics of bars by making the great simplifying ap proximation that the motion of stars normal to the galactic symmetry plane can be ignored The usual justi cation for this is summarized in 41 though it is now clear that this ap proximation is inadequate for bars the dynamical structure of a three dimensional bar is more than just a simple addition of independent vertical oscillations to orbits in the plane Neverthe less as many properties of orbits in three dimensional bars can be recognized as generalizations from two dimensions a preliminary discussion of the simpler but by no means straightforward two dimensional case remains an appropriate starting point As we have no analytic models for rapidly rotating bars even in two dimensions we might hope to construct an approximate one using Schwarzschild7s method 35 Despite the vast literature on two dimensional bars this programme has been carried through in only one case Pfenniger 1984b most papers con ne themselves to a discussion of orbits in arbitrarily selected bar like potentials having simple functional forms Clearly it is necessary to learn which of the many possible orbit families could contribute to a self consistent model and to understand how they are affected by changes in the potential a good grounding in both these aspects is an essential pre requisite for a search for a fully self consistent model It is also sensible to explore many types of model in the hope of nding some that might be more nearly integrable 34 However in two dimensions at least the eld is now suf ciently mature that such arguments are wearing thin 4 Thin disc appmm39matz39on There are three principal requirements which must be ful lled before three dimensional motion can be neglected as an unnecessary complication First the scale of the non axisymmetric Barred galam39es 17 structures7 spirals7 bars7 etc7 should be large compared with the thickness of the disc Second7 the oscillations of stars in the direction normal to the plane must not couple to motion in the plane Third7 there can be no instabilities which would cause the plane to warp or corrugate These three conditions are widely believed to be ful lled for nearly awisymmetrz39e discs The rst is doubtful only in the case of occulent spiral patterns The second seems valid because the frequency of z motion is higher than that of any perturbing forces which might arise from disturbances in the plane7 and therefore the motion normal to the plane should be adiabatically invariant see 51 Finally7 the principal instability which could give rise to corrugations of the plane7 the re hose instability7 is known to be suppressed by a modest degree of pressure normal to the plane7 and galaxy discs seem to be well clear of this stability boundary77 Toomre 1966 However7 the second and third requirements are violated for strong bars Combes and Sanders 19817 Pfenniger 1984a7 Raha et al 1991 and introduce the further aspects to the dynamics we discuss in 5 42 Linear theory With the exception of the Stackel models7 orbits in all strong bar like potentials must be cal culated numerically The bewildering variety of orbit types found in most numerical studies of a strong bar like potential is dif cult for the newcomer to assimilate Moreover7 the character istic diagrams and surfaces of section drawn in many papers7 require considerable explanation before the wealth of information they contain can be comprehended Before plunging into numerically calculated orbits7 therefore7 we rst explore linear theory for a weakly perturbed case Although a very crude approximation for bars7 the results give useful insight to those from more realistic strongly barred models We also introduce characteristic curves for orbits of arbitrary eccentricity in the simplest possible case of an amisymmetrie potential The equations of motion of a test particle eg a star or planet in the symmetry plane 73192 0 of a potential7 I7 are 7492 7 93 97quot 6lt1gt J 7 667 where a dot denotes a time derivative and J r2197 is the speci c angular momentum of a star It is useful to divide the potential into an unperturbed part7 which is axisymmetric7 and a non axisymmetric perturbation ltIgt ltIgt0 b In an axisymmetric potential a5 E 07 J is conserved and we de ne a home radius7 rC7 at which a star would pursue a circular orbit 6lt1gt0 J2 W To 737 where QC is the angular frequency of circular motion at re A star possessing more energy than the minimum required for circular motion for a given J7 oscillates about its home radius As the period of the radial oscillation is generally incommensurable with the orbital period7 the orbit does not form a closed gure in an inertial frame7 except for a few special cases such as harmonic or Keplerian potentials Successive apo galactica of the rosette like orbit are between 7139 and 27139 radians apart in all realistic galactic potentials 18 Sellwood and Wilkinson 42 Lindblad epicycles When the stars orbit is far from circular the radial motion is anharmonic and asymmetric about its home radius For nearly circular orbits on the other hand the stars motion approximates an harmonic epicycle about a guiding centre which orbits at Q as rst exploited by Lindblad 1927 in a galactic context To see this we describe the motion in terms of displacements g 77 from uniform circular motion g in the outward radial and 77 in the forward tangential directions Assuming the displacements g 77 to remain small the equations of motion 5 can be approximated as Hill 1878 doc 63 2m 7 2977 J dr 37quot 6 1 53 H 29c iiiv 77 g TC 30 where the potential derivatives on the right hand sides are to be evaluated along the circular orbit Setting the perturbing potential 75 0 for the moment the solutions to the homogeneous equations are 296 aem wo and 77 0761Ht 0 7 where a is the maximum radial excursion of the particle and the Lindblad epicyclic frequency is d9 12 C l lt8 These equations describe an elliptic epicycle about the guiding centre Since QQC gt H in all realistic potentials the major axis of the ellipse lies along the direction of rotation In linear theory we can treat the individual Fourier components of the potential perturba tion separately When a sinusoidal perturbation rotates at a uniform rate 9p we can write it as rt9t Preim9 npt where m is the azimuthal wavenumber of the perturbation m 2 is the fundamental wave for a bar or bi symmetric spiral The function P describes the radial variation of amplitude and phase of the perturbation for a spiral it is necessarily complex but it can be purely real at all radii for a bar As usual with complex notation the physical quantity corresponds to the real part With this form for 75 equations 7 become dQZ 3P C 7 if 7m99c79pt dr g 77 37quot 6 H 4937quot EH 7m 9 77 2965 7pemieltnrnpgtti Tc It is convenient to write the angular frequency at which the guiding centre overtakes the per turbation as w 7775C 7 9p The particular solution is 71 2 QC 3P g m P 51ltm9w 7 10839 H2 7 w 776w and 77 7in w 37quot w TC 2 2 2 72 2966713 4QC 7 F H T13 5W9 11b Barred galan39es 19 Angular frequency Figure 4 The angular frequency of circular motion QC full drawn curve and QC i nZ dashed curves for our arisyrnmtn39c mass model The horizontal dotted line shows the angular frequency of the frame adopted in 424 the intersections of this line with the dashed curves mark points where Very nearly circular orbits would form closed biisy39mmetiic gures in this frame which describes only the distorted path of the guiding centre The full solution which includes epicycles about the new distorted path is obtained by adding the complementary function the solution 8 to the homogeneous equations A resonant condition occurs wherever the denominators in equations 11 pass through zero This occurs at coirotation where w 0 and at the Lindblad resonances where w in The resonant denominators also imply that for a xed amplitude P the displacements increase without limit as the resonance is approached our linearized analysis must therefore break down near these radii Figure 4 shows the angular frequencies QC and QC j Km with m 2 as appropriate for a bar plotted as functions of radius in a reasonably realistic galaxy model The axisymmetric mass distribution used is described in 431 Resonances for nearly circular orbits occur wherever the horizontal line at the bar pattern speed 9p intersects one of these curves The pattern speed chosen for this illustration has one outer Lindblad resonance OLR and two inner Lindblad resonances ILRs in this mass model Clearly there would be no ILRs if 9p exceeded the maximum value of QC 7 K2 moreover this peak would be much lower if the galactic potential had a more extensive harmonic core ILRs need not be present in every barred galaxy therefore 422 Orbit mientatz39ons Equations 11 give the displacements of the guiding centre when subjected to a mild uniformly rotating sinusoidal potential perturbation The guiding centre 20 Sellwood and Wilkinson moves at a non uniform rate around a centred ellipse which is elongated either parallel or perpendicular to the potential minimum Sanders and Huntley 1976 In order to examine the sense of alignment more easily we neglect the radial derivative 61367 on the grounds that the perturbed potential is likely to vary only slowly with radius Dropping the exponential factor the radial displacement of the guiding centre is therefore approximately N 27719C 7 mum 7 to If we choose the line 6 0 for the bar major axis P is real and negative and g has the same sign as to between the Lindblad resonances but takes the opposite sign when lwl gt Is ie further from co rotation than the Lindblad resonances Thus the orientation of the orbit changes across every one of the principal resonances Between the ILR or the centre if none is present and co rotation the orbit is aligned with the bar but it is anti aligned beyond co rotation The orientation reverts to parallel alignment beyond the OLR The orbit therefore responds as a harmonic oscillator of natural frequency H driven at frequency u with a negative forcing frequency being interpreted as its absolute value but 7139 out of phase The switch of alignment across the Lindblad resonances occurs for the familiar reason that any driven harmonic oscillator is in phase with the forcing term when the driving frequency w is below its natural frequency Is but is exactly 7139 out of phase when the forcing frequency exceeds the natural frequency Even though equations 11 were derived assuming in nitesimal perturbations most studies in a variety of reasonable potentials have found that the more nearly circular periodic orbits are aligned with the bar inside co rotation and that stable perpendicularly oriented orbits are often found inside the ILR7 P 11 423 Action angle variables The epicyclic viewpoint remains valid for arbitrarily eccentric orbits in an axisymmetric potential but the motion can no longer be described by equations Kalnajs 1965 1971 was the rst to use action angle coordinates Jr Jawrwa to describe galactic orbits of arbitrary eccentricity In an axisymmetric potential the radial velocity of a star of energy per unit mass E and speci c angular momentum J is r2Eilt1gtr 7112 12 The period of its radial oscillation is 27139 quot1d r7 13 fr 7 739 9r and the frequency 9r thus de ned is constant for that star We de ne the angle wr to be the phase of this oscillation thus wr L The action conjugate to this angle known as the radial action is the area of the oscillation in phase space divided by 27139 in the normalization used in this eld ie 1 J i f Mr 14 r 2W 7 Contopoulos and Mertzanides 1977 found an example where a steep radial gradient in the perturbation which we neglected in equation 12 caused a local reversal of these normal orbit orientations B arrcd galartcs 21 I 39 I 39 I I 39 I I 39 I V I 39 I 39 I I 39 I 39 I 39 I 39 I I 39 I 39 I 39 I 39 I l I 39 I 39 I 39 1 I 39 I 39 I 39 I 39 I 0 O 1 1 2 1 31 4 1 5 1 lt gt s gt VIC 1 1 4 11114i 1 0 11 2 1 3t1 4 1 5 1 0 u Q van 0 d J 6 O a x gt quot 1 l n I A I n I A I J A I n l n l A l I A I A l n l A l 1 n l A l A I A l l n l u l n I A l l n I n l A I A I Figure 5 A single orbit drawn in frames rotating at many different rates The view marked 00 is from an inertial frame erame O and all the other views are seen when erame is given by equation 17 with the values of mzl marked in each The 10 frame shows the Lindblad epicycle In all frames with l negative the star orbits less rapidly than the frame whereas the reverse is true when l is positive The only frame which rotates counter to the star s orbit is that marked 11 The 100 dots in each panel are separated by equal time intervals for that frame The radial action has the dimensions of angular momentum and is a measure of the amplitude of a star s radial oscillation Because a star conserves its angular momentum during this radial oscillation it does not progress around the galactic centre at a uniform angular rate except for circular orbits but we introduce an angle 211 that does If A9 is the change in a star s azimuth during one radial period we define this angle to change at the uniform rate A6 211a E Q 15 Tr Thus 12 is the orbital frequency of the guiding centre and 211 its phase angle The action Ja conjugate to 211 is simply J the specific angular momentum For less eccentric orbits the motion tends towards the epicyclic description of 421 and these new coordinates behave as Jr a LZ Ja E J 16 Brae Qaa c For eccentric orbits in most reasonable potentials 12a and 121 are respectively less than Q and H evaluated at the guiding centre 424 Orbits in rotating frames Because bars and perhaps spirals are believed to be steadily rotating features it is sensible to study orbits in a frame which rotates with the non axisymmetric pattern A number of their properties in perturbed potentials can be understood more readily by first considering orbits in an axisymmetric potential viewed from rotating axes Although the orbit of a general star in an axisymmetric potential does not close when viewed from an inertial frame it will appear to close to an observer in a coordinate system which rotates at certain frequencies The angular rotation rate of the frame in which the orbit will appear to close must be erame Qa Lara m 22 Sellwood and Wilkinson Figure 6 Many orbits which all close in the same rotating frame Most orbits are highly eccentric but as the order of the symmetry rises the more nearly circular orbits lie progressively closer to the co rotation circle dashed for integer l and m Figure 5 shows the same orbit drawn in several rotating frames with the values of l and m marked for each The gure closes after the orbit has made m radial oscillations and l turns about the centre of the potential Since the frequencies vary from orbit to orbit most orbits will not close in one arbitrary rotating frame However as both S2 and S21 vary with both the speci c energy E and angular momentum J 18 can be satisfied for more than a few isolated orbits for a single Qfmme In fact there will be an infinite number of 1 D sequences in the two dimensional E J space along which the orbits appear to close To illustrate this important concept Figure 6 shows many different orbits which close in the same rotating frame These orbits were found numerically in the axisymmetric mass model described in 431 We have grouped these selected orbits by shape and it should be clear that each group represents a continuous sequence of orbits which close in this frame Obviously in this axisymmetric potential the orientation of the shapes is arbitrary we have simply chosen to draw them so that the axis of reflection symmetry is in the y direction Figure 7 shows characteristic curves for sequences of orbits in this potential which all close in the same frame The radial extrema of closed orbits are plotted as a function of the quantity EJ E J fmme with Qfmme held constant which is known as J acobi s integral see 432 and we plot these as negative values for stars with J lt O The heavily drawn line shows circular orbits which trivially close in any frame and the dashed zero velocity curve or ZVC marks the radii at which a particle would appear momentarily at rest in this frame These two curves touch at their maximum values of E J near the ordinate 7quot 125 at which a star on a circular B awed galamcs 23 Radial extremum Figure 7 Characteristic curves showing the loci of the radial extrema of families of orbits which all close in one rotating frame negative values are used for orbits which rotate in a sense counter to the frame The thick line marks the circular orbits as a function of the Jacobi integral and the dashed curve marks the ZVC which no orbit can cross Each characteristic curve is marked with the mzl value for which the orbits close orbit appears at rest zlc it co rotates with the frame All other lines on this Figure show parts of several characteristic curves Each is marked with the ratio of radial to orbital periods before the orbit closes in the rotating frame We have selected only orbits for which l tl or O and low values of m and the majority of sequences are incomplete only in the case of the 31 family have we drawn the sequence as far as almost radial orbits The 10 family are simply the extension of Lindblad epicycles 421 to more eccentric orbits Each mzil sequence crosses the circular orbits at a point known as a bifurcation In an axisymmetric potential each bifurcation occurs where eram ECOquot 517quot for the infinitesi mally eccentric orbits discussed above The 31 sequence crosses the circular orbit line at three points inside co rotation the bifurcation occurs only at the point where the orbits are nearly circular There is one bifurcation outside co rotation l 1 and one inside l 1 for all sequences except for the 21 orbits In this exceptional case the two sequences starting from the two 21 bifurcations are connected and form a closed loop around the circular orbits called 24 Sellwood and Wilkinson 06 quot39 04 quot 39 02 190 188 186 184 182 190 188 186 184 182 190 188 186 184 182 IJ quotE I J Figure 8 A schematic representation of the two principal types of resonance gaps The dashed curve in each panel is the ZVC the left hand panels show the bifurcation with no potential perturbation and the other illustrate the two different types of gap that a perturbation can create These diagrams are drawn for the situation inside co rotation a bubble by Contopoulos 1983a The 11 bifurcations occur outside the diagram the l 1 lies at large EJ for retrograde orbits and the l 1 at large 7 The two branches of each sequence on either side of the bifurcation are not independent since each traces one of the two extremal radii for the same orbit Each sequence also touches the ZVC where the orbit develops cusps More eccentric orbits loop back on themselves for part of the time Figure 6 but we have drawn this continuation only for the 31 family in order not to clutter the diagram with too many crossing lines Similar sequences of higher m still with l 1 can be found closer to co rotation and are ever more densely packed as they approach this point More sequences occur for higher values of l and correspond to orbits which close after l rotations For small m they are found either far outside co rotation or are retrograde if they exist at all but the higher m sequences could also be drawn within the boundaries of Figure 7 in fact sequences for all ml would densely ll the plane except for the region excluded by the ZVC 425 E ect of a weak nonarisymmetm c perturbation If we again add a weak perturbing potential co rotating with this frame 216 so erame Qp all these closed non circular orbits resonate with the perturbation We should therefore expect them to be substantially altered in a perturbed system We have already calculated the effect on the circular orbits in 422 An in nitesimal pure m fold symmetric potential perturbation introduces Lindblad resonances at the mzi1 bifurcations No closed orbits can exist at the resonant point because a particle there would experience a monotonic acceleration until it is driven off resonance Linear theory cannot tell us any more but exact calculations show that the four branches at a bifurcation separate into two non crossing pairs the resonance imposes a gap both in the circular orbits and in the 77121 sequence The stronger the potential perturbation the wider the gap becomes We can see which pairs must join from the direction in which the perturbation moves the circular orbits in the characteristic diagram Figure 7 a small part of which is shown enlarged in the rst panel of Figure 8 It is customary in these diagrams to plot the intercept of the Barred galam39es 25 distorted orbit on the minor axis of the potential perturbation thus the circular orbits are displaced in the direction given by the sign of 75 in Ha When 61367 is negligible g has the same sign as QMQCTcw 2 from since P is negative In this case the circular orbit branch on the co rotation side of the Lindblad resonance bends away from the ZVC while that on the other side of the resonance bends towards the ZVC which is called a gap of type 1 On the other hand should the perturbation amplitude decrease rapidly with increasing radius the term containing 61367 could outweigh the other in Ha under these circumstances the bends occur in the opposite sense creating a gap of type 2 Both types of gap are shown schematically in Figure 88 Linear theory predicts gaps only at the Lindblad mil resonances Finite perturbations however give rise to many more gaps for two distinct reasons Firstly in most reasonable bar like potentials the perturbation is not a pure m 2 sinusoid but also contains other Fourier components We should therefore expect gaps at every mil resonance for which the Fourier component m is non zero these will be at even values of m whenever the perturbation has exact two fold rotational symmetry However gaps continue to occur at all even m bifurcations when the perturbation has a pure cos2t9 form Contopoulos 1983a This is because the nite radial extent of the distorted path of the guiding centre breaks the precise m fold symmetry of the potential explored by the star The rst such additional resonances for non linear orbits the 2mi1 resonances are known as ultra or hyper harmonic resonances and occur at 4amp1 for a bar No gaps develop at the odd mil bifurcations for a purely symmetric potential perturba tion Some authors rather confusingly continue to refer to a resonance gap as a bifurcation the phrase pitchfork bifurcation is useful to emphasize a bifurcation without a gap 43 Strong bars The daunting task of summarizing the major results on the dynamical structure of two dimen sional stellar bars is made still harder by the number of rival conventions adopted by the various groups The most obvious is the number of different naming conventions for the orbit families in current use but even such simple things as the bar orientation in diagrams has not been standardized 69 its major axis lies up the page in some papers and across the page in others and a recent author has decided it looks nicer at 45quot While a careful reader can always infer the orientation it is frequently left unstated Other inconsistencies abound in the important characteristic diagrams 44 A large number of different mass models has been investigated This has the dual advantages that we can separate what is generic from what is accidental77 Contopoulos and Grosbol 1989 and can begin to understand how the properties of the orbits change as parameters are varied However this strategy makes it impossible for a reviewer to select sample diagrams from the literature to illustrate all aspects of a single model The relation between the three principal diagrams showing shapes of periodic orbits characteristic curves and surfaces of section and the reason for constructing all three is much easier to comprehend if an illustrative example of each can be shown for the same model 8 This is not precisely the convention used by Contopoulos and Grosbol 1989 who de ne 4 types of gap 26 Sellwood and Wilkinson 43 Mass model We have therefore anchored this part of our review around a single mass model which we introduce to illustrate the main results from the literature The model we have selected is constructed from building blocks in the usual arti cial manner However it is among the simplest which display many of the properties frequently discussed and its orbital structure is qualitatively similar to the more realistic N body and photometric models discussed in 492 amp 493 for which the gravitational potential is known only numerically We also make no attempt to ascertain whether the orbital structure of our model is favourable for complete self consistency Our model which most closely resembles those used by Sanders and his collaborators has the following three mass components B A uniformly tumbling Ferrers 1877 ellipsoid to model the bar This is an inhomogeneous prolate spheroid of mass MB having a density pro le 105 M 1 7 2 2 lt 1 M3077 SZWIZCZ B M j gt 1 where 2 2 2 z y 2 2 g CZ 7 19 and a gt c The major axis of the ellipsoid a is therefore aligned with the s axis following the convention in Binney and Tremaine 1987 and the bar and coordinate system rotate about the z axis De Vaucouleurs and Freeman 1972 give the potential of this mass distribution in the following convenient form let 62 a2 7 02 and de ne 7 by y2 tan2 7 x2 sin2 7 62 M gt 1 20 cos7 067 M S 1 1 wzkw 2 tan21 16sin2k 16d6 0 These integrals are all straightforward The potential can now be written as 7 105GMB I BWW 32E 1 1 gwio glt2w11 73121020 1 glt4w12 2292w21 73141030 21 1 ig6w13 3x4y2w22 32y4w31 y6w40 U A small dense spherically symmetric component to model the bulgespheroid which has the density pro le of a Plummer sphere i 2 pm 7 Mg 1 T2gt7 22 47Ts3 52 where Ms is the mass of the spheroid and s is a length scale Of course 7 is here a spherical radius Barred galaxies 27 a a adm mm A Radius Radius Figure 9 The two force components at various azimuthal angles in our barred model as functions of radius The boundaries to the shaded areas indicate the largest and smallest values attained around at circle at that radiusi Within radius 1 the semiemajor axis of the bar the forces vary strongly with azimuth but their range decays rapidly outside the bar H An extensive spherically symmetric component to model the halo which has the same form as equation 23 but a much lower central density larger total mass MH and longer length scale hi Although all three components are 3dimensional we require the potential in the plane 2 0 only 7 its variation out of the plane is of no importance in this section We therefore do not need to distinguish a separate disc component component H can be thought of as representing both the disc and halo The advantages of choosing a Ferrers ellipsoid to represent the bar are that the potential and its derivatives can be written in closed form and that the potential is continuous up to its third derivative because only the second derivative of the density is discontinuous at the boundary The disadvantage is that being elliptical with a constant eccentricity at all radii the mass distribution does not resemble that of real bars see 2i3i We have adopted this model here because it is easy to program and widely used and because it exhibits almost all the generic properties we wish to illustrate We set the gravitational constant G 1 and adopt the bar major axis as our unit of length a 1 in this unit 5 0 05 and h 15 The masses of the three components are MB 11852 Ms 0 3 and MH 25 The axis ratio of the ellipsoid we choose to be 10 3 and we set its angular rotation rate 9p 2 in these units The axisymmetric mass model used in 4i2 was very similar differing only in that the Ferrers ellipsoid was spherical ac 1 and had a mass of precisely 1 unit Figure 9 shows the azimuthal variations in both the radial and tangential forces in our model The gravitational forces therefore vary quite strongly with azimuth within the bar but quickly approach axial symmetry beyond it 28 Sellwood and Wilkinson 0 Figure 10 Contours of the effective potential in our barred model and the locations of the ve Lagrange points The L3 point at the centre is a minimum the points marked L4 and L5 are equal absolute maxima and those marked L1 and L2 are saddle points Well beyond this ring of Lagrange points the effective potential slopes away parabolically to in nity The bar major aXis is across the page 432 Rotating nonazisgmmetric potentials The gravitational potential of a uniformly rotating non axisymmetric density distribution is steady when viewed from axes which co rotate with the perturbation We de ne an e ective potential in a frame rotating at the angular rate Qp Pfo I QI27 2 where 7 is the distance from the rotation centre The equation of motion in the rotating frame of reference may then be written a vcigte 2op X 739 24 Though the second Coriolis term complicates the dynamics the gradient of the effective potential at least determines the acceleration of a particle momentarily at rest in this frame Figure 10 shows contours of the effective potential in our model The topology has been aptly likened by Prendergast 1983 to that of a volcano there is a deep minimum at the centre which forms the crater a rim whose height varies slightly and steep walls sloping away to in nity There are ve Lagrange points altogether at which the gradient of Peg is zero a Barred galaries 29 minimum at the centre L3 and the other four lie at symmetry points on the crater rim there are two global maxima on the bar minor axis L4 and L5 and two saddle points on the major axis L1 and L2 A stationary particle located at any one of these points would remain at rest in this rotating frame or would co rotate with the bar The equilibrium at the saddle points L1 and L2 is always unstable that at the bar centre L3 is always stable but the stability of the minor axis Lagrange points L4 and L5 depends upon the details of the mass distribution eg Binney and Tremaine 19879 they are stable in many reasonable models including that we have chosen Neither the speci c energy E nor the angular momentum J of a particle is separately conserved in a rotating non axisymmetric potential but Jacobi7s integral EJ E 7 QpJ introduced in 424 is conserved It may also be expressed in terms of the effective potential EJ lflz Dem 25 and is therefore the energy with respect to rotating axes As a consequence it is often loosely referred to as the energy This form for EJ renders the concept of the effective potential doubly useful since it deter mines whether a particle of a given energy EJ is con ned to particular regions of the space A particle for which EJ lt ltIgteg at the Lagrange points L1 or L2 is con ned to remain either inside co rotation or outside it Only those particles having EJ gt ltIgteg at the Lagrange points L4 or L5 the absolute maxima of 53 are free energetically to explore all space Note however that a particle having a speci c EJ will not necessarily explore the whole region accessible to it in particular particles beyond the Lagrange points may not be unbound in reality even though they appear to be so energetically because the Coriolis force generally prevents them from escaping to in nity 44 Periodic orbits A simple periodic orbit is a special orbit which a star would retrace identically on each passage around the galaxy in the rotating frame of the perturbation More complicated periodic orbits also exist which close after more than one passage around the galaxy All orbits which close in a steady rotating potential are therefore periodic orbits 424 Stars in any barred galaxy are most unlikely to follow periodic orbits yet the study of periodic orbits is of interest because many non periodic orbits are trapped to oscillate about a parent periodic orbit in a manner exactly analogous to non circular orbits following the path of the guiding centre which is a periodic orbit in an unperturbed or weakly non axisymmetric potential 421 A periodic orbit therefore gives an approximate indication of the shape of the density distribution of stars trapped about it Unlike the in nitesimally perturbed case however it is possible to nd orbits which are not trapped about any periodic orbit but explore a much larger region of phase space This is the distinguishing property of a near integrable system 34 We may determine whether an orbit is trapped or not through an examination of the surface of sectiori 46 introduced by Poincare 1892 9 Unfortunately their application to the logarithmic potential has a minor aw pointed out by Pfenniger 1990 30 Scllwood and Wilkinson 44 Techniques The shooting method is the most widely used to nd periodic orbits One chooses an initial set of coordinates in phase space X0 usually on one of the principal axes of the potential and integrates the orbit until it recrosses the same axis at some point XI The orbit integration can be considered as a mapping Tz0 131 and a periodic orbit is then de ned by TWO 0 The map is in fact a Hamiltonian map cg Lichtenberg and Lieberman 1983 and periodic orbits are also known as cccd points in the map To nd such a point we calculate the mapped points for other trial orbits in the vicinity of an initial guess X0 to determine the changes required to x0 in order to move the end point x1 towards x0 The calculation proceeds iteratively and generally converges quickly to a periodic orbit to high precision An alternative relaxation or Henyey method is described by Baker ct al 1971 but is less widely used Although x0 has four components for planar motion there are only two degrees of freedom for the search since the Jacobi integral is conserved by the mapping and a change of starting phase along the orbit is trivial It is customary to search for orbits which cross the minor y axis of the bar with y 0 by adjusting yo and i0 All orbits found by this search strategy must be re ection symmetric about the y axis searches in which i0 and ye are varied are required to nd asymmetric periodic orbits 442 Orbit stability A star on any periodic orbit will forever retrace exactly the same path Floquet7s theorem cg Mathews and Walker 1970 Binney and Tremaine 1987 tells us that a star whose starting point is very close to that of a periodic orbit will follow a path which either winds tightly around that of the periodic orbit or diverges away from it in an exponential fashion at least while the departure from the periodic orbit remains small These two types of behaviour imply that the periodic orbit is respectively stable or unstable In practice having found a periodic orbit one then integrates two neighbouring orbits each having the same value of EJ but displaced slightly from x0 in di ferent directions to nd the end points x1 where it recrosses the y axis In the neighbourhood of a periodic orbit these must be approximately 9T 21 Tz0 62 2 0 7 6x 27 62 to The 2 gtlt 2 Jacobian matrix of partial derivatives of T can be estimated numerically from the displacements of the nal point given the initial displacements 6x Because Hamiltonian maps are area preserving cg Lichtenberg and Lieberman 1983 the determinant of the matrix and therefore the product of the eigenvalues is unity Since the matrix is real there are just two cases the orbit is stable if the eigenvalues are complex and lie on the unit circle and it is unstable when the eigenvalues are real and one lies outside the unit circle The two cases are respectively elliptic or hyperbolic xed points in mapping parlance 45 Pcriodic orbit familics Numerous orbit families have been described in the literature We do not attempt to enumerate them all here but con ne our discussion to those which are relevant to the structure of real bars Contopoulos and Grosbcl 1989 give a more comprehensive review Barred galam39es 31 A selection of periodic orbits supported by our model 431 is shown in Figure 11 In every panel of this gure7 the dashed curve marks the outer boundary of the Ferrers ellipsoid Stable periodic orbits are full drawn7 unstable ones are dotted There are a number of different naming conventions for the orbit families In Figure 11 we use that developed by Contopoulos7 which is neither descriptive nor easily memorable7 simply because it is the most widely used Athanassoula et al 1983 attempted an alternative system but7 like others7 theirs is not easily generalized to three dimensions7 and even those authors have reverted to the Contopoulos system in their more recent papers The 1 family Figure 11a7 also referred to as the main family7 is elongated parallel to the bar within co rotation The 2 family is represented by the two orbits within the bar in Figure 11b note that these are elongated perpendicularly to the bar axis We do not show any x3 orbits which are about as extensive as the 2 family7 but are much more elongated and always unstable The retrograde x4 family Figure 11c are very nearly round7 but are slightly extended perpendicular to the bar All four xi families are 21 orbits Figure 11d e shows both the inner and outer 31 and 41 resonant families while Figure 11f shows the short and long period orbit families which circulate about the Lagrange points We show characteristic curves for most of the main periodic orbit families in Figure 12 This diagram7 which plots the y minor axis intercepts of the orbits in each family plotted as a function of Jacobi7s integral7 should be compared with that for the axisymmetric potential Figure 7 Again we differentiate between stable full drawn and unstable dotted families 711 all bound orbits in an axisymmetric potential are stable Figure 7 There are broad similarities and important differences between Figures 12 and 7 The most obvious difference is that the circular orbit sequence in the axisymmetric potential has been broken by numerous gaps in this strong bar case these gaps are so large and frequent in some regions that the sequence becomes hard to trace7 especially inside co rotation 45 Bi symmetm39c families Starting with retrograde orbits7 we see that the x4 family in Figure 12 closely corresponds to the retrograde circular orbits in Figure 7 Both the charac teristic curve and the orbit shape Figure 11c indicate that these counter rotating orbits are not much affected by the bar potential7 being only slightly elongated perpendicularly to the major axis These seem to be generic properties of the retrograde orbits in every potential investigated The direct 21 families are substantially affected by the bar however7 those beyond co rotation are more easily related to their counterparts in Figure 7 The orbital eccentricity of the bi symmetric families outside co rotation rises in the vicinity of the OLR The orbits of the outer nearly circular family are elongated parallel to the bar Figure 11a while the inner family are perpendicular7 creating a typical gap of type 1 All this is consistent with the linear theory predictions of 427 even though the eccentricity rises well beyond the linear regime The correspondence with Figure 7 is less apparent inside co rotation7 because what was the circular orbit sequence has been broken by two yawning gaps For EJ lt 718 the nearly circular orbits have become very elongated and the characteristic curve is therefore much further from the ZVC At the 21 resonant gap7 the nearly circular sequence joins to the lower branch of the bubble in Figure 7 while the continuation of the circular sequence has broken away and lies much closer to the ZVC to form the 2 sequence The third branch7 that which lies very close to the ZVC in Figure 77 becomes the highly eccentric and unstable 3 family Thus the 1 family derives from the circular orbits only very near the centre and near EJ 720 while 32 Scllwood and Wilkinson 39 I I l I l I l I I I l 1 I 2 o 2 2 o 2 2 o 2 Figure 11 Some periodic orbits in our barred model those drawn with full lines are stable orbits unstable orbits are drawn as dotted curves The boundary of the elliptical bar is marked with a dashed curve Orbits from the main 1 family are shown in a The small orbits near the bar centre in b are 2 while the others are members of the outer 21 families c shows the retrograde family 4 d 85 e show inner and outer 31 and 41 orbits respectively and f shows two kinds of orbits about the Lagrange points the elongated banana orbits are known as long period orbits LPO while the rounder are members of the short period orbit SPO family nb both SP0 and LPG orbits occur around both Lagrange points at intermediate energies it derives from the eccentric 21 family Contopoulos 1988a calls the closed loop formed by the 32 and 173 families a floating bubble The appearance of Figure 12 is typical of a strong bar the resemblance to Figure 7 is much closer for weaker bars The existence of an ILR in the axially symmetrized model is a necessary but not su icz cnt condition for the perpendicular 12 and 123 families to exist They are therefore absent when the central density is less high Teuben and Sanders 1985 or when the pattern speed is higher However studies in which the bar strength is varied show that the gaps at the two 21 bifurcations in the axisymmetric case widen and the radial extent of the perpendicular families shrinks as the bar strengthens Eventually the 12 and 123 families completely disappear for very strong bars Contopoulos and Fapayannopoulos 1980 Van Albada and Sanders 1983 suggest that the existence and extent of the 172 family can be used to generalize the concept of an ILR to nite amplitude perturbations As also found by Athanassoula ct al 1983 the 321 family in our model does not have loops at higher energies Figure 11a However many other papers Contopoulos 1978 Pa payannopoulos and Fetrou 1988 Teuben and Sanders 1985 have reported 321 orbits with loops Barred galaries 33 Figure 12 Characteristic curves for orbits in our bar model The point at which the orbit crosses the bar minor y aXis is plotted as a function of Eg for each family of periodic orbits Full drawn segments of these curves indicate the orbit family is stable and unstable parts of the sequence are marked as dotted lines The dashed curve shows the ZVC and the three vertical dot dashed lines mark the energies at which the surfaces of section are drawn in Figure 13 This diagram should be compared to gure 7 which are sometimes very large Athanassoula 1992a shows that such loops appear when the bar either rotates more slowly or has a high axial ratio or when the mass distribution is more centrally concentrated 452 Higher order resonance families Because the potential in our model is bi symmetric the mzil bifurcations in the axisymmetric case become gaps only for even m Although it is hardly noticeable in Figure 12 the 81 family becomes unstable close to the odd bifurcations a discussion of this phenomenon is given by Contopoulos 1983a 34 Sellwood arid Wilkinson Extensive 31 families exist both inside and outside co rotation Figure 11d It is very common for the inner 31 family to be unstable but few authors report that the sequence becomes stable again for very high eccentricities Papayannopoulos and Petrou 1983 found a family of orbits having a similar appearance but which in their case resulted from a 11 bifurcation in a rather slowly rotating bar Our model rotates too rapidly to exhibit a 11 bifurcation the conditions under which this could be present are discussed by Martinet 1984 The gap in the characteristic curves at the inner 41 resonance is particularly wide even for our ellipsoidal bar square ended bars in real galaxies can be expected to have still stronger in 4 components to the potential and therefore stronger resonances and yet bigger gaps Notice that the gap in this case is of type 2 which means that the low energy 1 family joins the more nearly rectangular orbits of the 41 family As orbits on this branch remain aligned with the bar while orbits on the other branch are roughly diamond shape and anti aligned the gap type at this resonance may be important for self consistency The type of gap to be expected is more dif cult to predict at these higher order resonances as it depends upon the relative amplitude and radial variation of more than a single Fourier component Contopoulos 1988 Type 2 gaps seem to be more common eg Athanassoula 1992a Despite the strong resonance the qualitative appearance of the 41 orbits Figure 11e particularly further out is not greatly affected by the presence of the bar of Figure 6c As we move closer towards the Lagrange points the pattern of resonant gaps for even families and pitchfork bifurcations for odd recurs repeatedly The additional families are ever more closely con ned to the vicinity of co rotation and of rapidly vanishing importance to a self consistent bar 453 Orbits arourid the Lagrange points The two families of orbits shown in Figure 11f are present whenever the Lagrange points L4 and L5 are stable note that both families exist about both Lagrange points The more energetic short period orbits SPO derive simply from Lindblad epicycles 421 in the unperturbed potential The banana shaped long period orbits LPO are the extension of the nearly circular sequence 1 which continues through all the bifurcations as the Lagrange point is approached Papayannopoulos 1979 Once the energy exceeds the effective potential at the major axis Lagrange points L1 and L2 a nearly circular orbit crossing the minor axis cannot complete an orbit around the entire galaxy but will be turned back as it approaches the bar major axis The LPOs are of exactly the same type as the horseshoe orbits discussed in the eld of planetary ring dynamics eg Goldreich and Tremaine 1982 Because they lie at large distances on the minor axis neither of these families has any relevance to self consistent bar models though the LPO family may be important for rings V 454 Other types of periodic orbit All orbits shown in Figure 11 are re ection symmetric about the minor axis and cross this axis with y 0 Yet more families of asymmetric periodic orbits exist some of which are stable They are generally ignored in the majority of papers since stars following asymmetric families are not expected to be present in large numbers in symmetric bar models Moreover we have described only simple periodic orbits which return to the same point on the bar minor axis at each crossing Many more periodic orbits exist for which the star returns to the same point only after more than one orbit around the galaxy Again these are not Barred galaries 35 thought to be present in great numbers in any bar model but the existence of these additional families is very important for the onset of chaos see 47 4 6 Nari periodic orbits The orbits of most stars in a bar are not periodic but when most are trapped to librate about a parent periodic orbit the structure of the bar is largely determined by the shapes of the parent orbits It is important therefore to determine the extent to which orbits are trapped A stable periodic orbit must support a trapped region but the extent of this region cannot be determined from the stability test 442 conversely an unstable orbit has no orbits trapped about it but could lie in a region where all nearby orbits are trapped about stable periodic orbits The stability test alone therefore does not tell us the extent of the trapped region and we need the more powerful tool of the surface of section SOS to determine whether one trajectory through 4 D phase space oscillates about another We illustrate this tool with three examples in Figure 13 again drawn for our model Note that the scales differ substantially between the three panels and that for the two lower energies we have not drawn what happens outside co rotation To construct these diagrams we integrate the trajectory of a test particle numerically and mark the point in the plane each time it crosses the line z 0 The points created by successive crossings of the y axis are called carisequerits Only points for which i lt 0 are plotted thus points with y gt 0 are created by a star whose orbital motion is in the direct sense in the rotating frame This is the same Hamiltonian map discussed in 441 Each panel in Figure 13 contains consequents calculated for many different orbits all having the same Jacobi constant the dashed curves mark the boundaries of the region accessible to any orbit of that energy A number of different types of behaviour can be seen 46 Regular orbits The single most striking feature in all three panels of Figure 13 is that in some parts of the plane the consequents lie on closed curves known as invariant curves an orbit which gives rise to an invariant curve is known as a regular or quasi periodic orbit Successive consequents are usually well separated along each curve but as the orbit is followed for longer and longer the consequents populate the curve more and more densely Topologically the star is con ned to a two dimensional toroidal surface in the 4 D phase space and the closed invariant curve in the surface of section is simply a cross section of this torus lnvariant curves from different orbits of the same energy can never cross l0 and characteristically form a nested sequence generally centred on the single point representing the parent periodic orbit or elliptic xed point It is clear from Figure 13 that the trapped region around some periodic orbits covers a substantial region of accessible phase space At EJ 723 the regular region on the right is trapped about the family 2 that near the centre about 1 and the closed curves on the left are trapped about a retrograde orbit from family 4 10 An invariant curve appears to cross itself at an unstable periodic orbit ag H non and Heiles 1964 but the curves there are the limits of two hyperbolae whose apices just touch at that point Sellwood and Wilkinson x quotcg 39 32quot u 39 2 I quot 939 I I 39 x f 3 2 3 395 39 39 4 n 39 l i l39 0quot i a o z 39 39 a 39 39 39 quot 39 rquot 39 I quot 3 I Ia oquot 1quot 39 39 s 39 N 39 v quot 3 w l n 39 I V1 39 n39 39 o 39 2 39 039 39o u i I 3 o n f 39 1 g 39 lL tz 39 39 039quot h 39 5 4 v o 39 I a Q quot 2 l 39 39 l quotI a s I y quot 3 I 1 390 a fw I g quot g 3 l 1 t 39 1 c o 39 39 39 39 39 39 39 l 39 I I I o 39 z a I 39 Q c l I g t1 r nu E quot gt o L i O u 1 z i O l 39 sswt 5 i o l gt l 39 quot R 1 o 39 I 39 a x I 3 t h l Lquot 5 c I 391 39 39 o u g 39 39 39 o a a MM 39 39u 3 K u v I 39 39 39 w quot quot 39 39 39 u 0 39 39I t 39 0 39 39Vf39 39 w a a 39 39o I h I I c o39 u quot17 quot M I 139quot 39 39 if 39 l quot s d I 39 0 A 0 39 39 ll39 Q n 39r I k 39 xc39 uoquot amp f a a 39 539 lbquot Aquot quot7 quot39 39 39 N a J t s z 2 quot Jw v39 39 39 u quot 57 quotquot I I l l3939 l I EJ 18 2 n 2 r I i q a 39 39 n 39 39393939 l quot 393 o 39 a z I I 3Kquot E I i 39 1 f s o 39 I r u C gt A I quot I 39 quot 3 f quot39 quotquot oquot 39 I H I 39 s quotl W r I39 I t o t 39 quot39quot 339 Iquot Warm I 0 g39mlgonlu i 39 JI139 quot 39 quot5 39 s 1 5 n39 q39 y 3 21 39 39 rquot g f 39 5 3939397 umwquot i i O I 75 a J l 2 quotquotquot 3939 quot39 39 39 6 I 39 Q V y O 393 f quot5 quot 39 r quot39i a suaquotquotquotquotquot a 539 Q quot I 2 5 a39 3 5quot a 39 t r39 q I U f h 1 gt E 39 I 3 394 i 39 39 i vhf gt o 45 l5 H gt g t 4 39 I 2 wquot393is39 k rquot I 1 Jl 2 3 Figure 13 Surfaces of section at three energies in our model many different orbits of the same energy contribute to each plot The points mark the y values at which an orbit crosses the y aXis with i lt O and more such crossings are marked for complex orbits The dashed curves surround the region to which all particles are con ned energetically at lower energies top two plots we show only the region inside co rotation but the energy of particles in the lower plot is suf cient for them to cross the Lagrange points Barred galaries 37 462 Irregular orbits Not every orbit gives rise to an invariant curve7 however Well beyond these three regular regions for EJ 723 is an example of an irregular orbitM The consequents from an irregular orbit gradually ll an area of the plane more and more densely with a random scatter of points as the integration is continued In this case7 the area is bounded by the zero velocity curve and the regions occupied by regular orbits As can clearly be seen for EJ 718 in Figure 137 an irregular orbit may also be con ned between two closed invariant curves This is an example of a semi trapped orbit MacKay et al 1984 note that some irregular orbits appear to be con ned to a part of the surface of section for a very large number of periods7 before suddenly crossing to spend a long time in another part The boundary7 or earitorus Percival 19797 which appears to separate these regions may be likened to an invariant curve that is not water tight We have not found a very clear example of this behaviour in our model 463 Other features In the surface at EJ 723 just outside the fourth invariant curve around 1 there are two examples of a more complicated invariant curve one has three distinct loops7 the other seven This type of invariant curve is created by a regular orbit trapped about a periodic orbit which closes only after several crossings of this plane A further example occurs around the retrograde family7 but in that case the orbit closes after two passages and is also asymmetric We have not drawn the twin of this orbit7 its re ection about the y axis7 since it would no longer have been clear that these were a pair of distinct periodic orbits The nal type of behaviour of note is to be seen surrounding the seven fold regular orbit This orbit illustrates a dissolving invariant curve the consequents surround the chain of islands7 and although the orbit is clearly not regular7 it does not escape to the stochastic region just outside it for many crossings This is a characteristic feature in the SOS when the degree of regularity is changing with energy H non and Heiles 1964 at energies just a little lower EJ 724 phase space is entirely regular7 while at slightly higher energies EJ 722 the trapped region around 1 is very small 464 Higher eriergies In the SOS at EJ 718 the regular region on the directly rotating side has recovered somewhat7 but it is centred around the 41 orbit near y 057 which is well outside the bar The bar semi minor axis is The regular region around the retrograde x4 family is still strongly evident7 but the energy is now too great for the 2 family to be present At EJ 717 the orbits are unconstrained energetically Phase space near the Lagrange points is still regular7 but some invariant curves appear as segments at both positive and negative y because7 as it oscillates about the parent periodic orbit7 the orbit may cross the minor axis going in either direction on either side of the galaxy As such orbits spend most of their time on the minor axis well outside the bar they cannot be of any real importance to the maintenance of the bar density 47 Oriset of chaos The reason for the dissolution of invariant curves is one of the principal concerns of non linear dynamics Lichtenberg amp Lieberman 1983 and H non 1983 give accounts of this complex 11 In astronomical papers7 these orbits are also variously denoted as chautie7 stochastic or even semiergadie7 but these terms are not all considered to be fully interchangeable in other fie si 38 Sellwood and Wilkinson issue but we have also found the article by Dragt and Finn 1976 and the informal review by Berry 1978 especially helpful Unstable periodic orbits provide the key Although most lie in irregular regions of phase space not all of them do 7 examples are in the original H non and Heiles experiment and our model discussed here Unstable periodic orbits can be found in regular regions when the two branches of the invariant curve from that point meet again only at another unstable periodic orbit forming a separatmm The region around the xed point suddenly becomes ergodic as soon as two branches from hyperbolic xed points fail to join smoothly but cross at some point other than a xed point The breakdown of large scale regularity seems to take place in stages however as chains of islands of stability survive for short energy ranges around rnultiply periodic stable orbits These chains are often surrounded by dissolving orbits as in Figure 13 signifying the breakdown of the corresponding separatrix The KAM Kolrnogorov Arnold and Moser 69 Moser 1973 theorern concerns the survival of invariant tori when an integrable system such as an axisyrnrnetric disc is subjected to a perturbation such as a bar The theorem states that those tori suf ciently far from resonance77 survive in a deformed state when a suf ciently small77 perturbation is imposed It does not predict the strength of the perturbation suf cient to destroy a torus but it does give a scaling law related to the order of the resonance concerned and the associated frequencies Thus we should expect as we have found that trapped orbits exist in non axisyrnrnetric potentials We also observe that the extent of the regular region of phase space generally dirninishes as the strength of the bar perturbation rises This last observation is particularly true near co rotation where ever more resonances occur as co rotation is approached As the strength of the perturbation rises more and more of these resonances overlap which is a condition for the destruction of regularity Chirikov 1979 Athanassoula 1990 has argued that the square ended nature of the bar density distribution may also hasten the onset of chaos in this region because the m 4 component of the potential has greater strength relative to the m 2 than would a more elliptical potential causing greater overlap of resonances for a xed bar strength Further Martinet and Udry 1990 note that interactions of higher order resonances in the vicinity of the unstable x3 farnily seem to be a particularly effective generator of chaos They argue that the contraction of this family as the angular speed of the bar is raised may account for the apparent reduction in the chaotic fraction of phase space found in faster bars Contopoulos 1983a noted that an in nite sequence of period doubling bifurcations in the characteristic diagram is a second factor which gives rise to ergodic behaviour This seemed to occur over a wide region around co rotation in his strongly barred case Contopoulos 1983b Liapunov exponents Liapunov 1907 which give a quantitative measure of the degree of stochasticity in an irregular region have been calculated in recent studies 69 Udry and Pfenniger 1988 Contopoulos and Barbanis 1989 They are a set of exponents describing the rate of separation of two nearby orbits in phase space as the motion proceeds the region is regular if all the exponents vanish irregular otherwise They are formally de ned for orbits in nitely extended in time but in astronornical systems the interest lies in the behaviour over a Hubble time 48 Actions ln steady potentials one quantity is conserved for all orbits the energy if the potential is non rotating or Jacobi7s integral in a non axisyrnrnetric rotating potential If this were the only Barred galam39es 39 conserved quantity the particle would be able to explore all parts of phase space accessible with this energy and the entire surface of section would be lled by one single irregular orbit We already noted that regular orbits are however con ned to a two dimensional toroidal surface and therefore respect an additional integral other than the energy In an axisymmetric potential this additional conserved quantity is obviously the angular momentum but no such simple physical quantity can be identi ed in non axisymmetric potentials whether stationary or rotating Binney and Spergel 1982 1984 give a vivid illustration that regular orbits have just two independent oscillation frequencies and can therefore be described by action angle variables lrregular orbits on the other hand are not quasi periodic and cannot be described by such variables Their rst application was for a planar non rotating potential but the technique is not restricted to this simple case Since the actions are a set of integrals they would furnish the ideal variables with which to describe regular regions in as much detail as for integrable systems such as Stackel models 33 We could write the dynamical equations in a very simple form 69 92 express the distribution function in terms of the actions Jeans theorem and take advantage of their adiabatic invariance away from resonances for the study of slowly evolving models Their principal drawback however is that we have no analytic expressions for them though they can be determined numerically from the area bounded by the invariant curve in the appropriate surface of section 69 Binney et al 1985 Worse we cannot easily transform back to real space coordinates in order for example to compute the shape of an orbit of known actions This very serious limitation is a major handicap to progress in the entire subject Ratcli f et al 1984 suggested a general technique based upon Fourier expansion but ran into a number of operational dif culties The canonical mapping approach outlined by McGill and Binney 1990 may be more successful 49 SeU consz39stency It is widely believed that self consistent bars are largely supported by stars on orbits trapped or semi trapped about the 1 family which is highly elongated in the direction of the bar Most other families are too round or elongated in the opposite sense to make any useful contribution in a self consistent model Somewhat surprisingly the 41 family does not appear to be responsible for the rectangular shapes of real bars 231 Contopoulos 1980 was the rst to argue that the properties of the 1 family suggest that self consistent bars were likely to extend almost as far as co rotation Teuben and Sanders 1985 concluded that stars in such rapidly rotating bars are likely to move in a well organized streaming pattern much as observed cf 25 whereas much less coherent streaming would be expected were the bar to rotate more slowly Petrou and Papayannopoulos 1986 argued that self consistent models having much lower pattern speeds might also be possible However the majority of authors favour pattern speeds high enough that co rotation is not far beyond the end of the bar We have already noted that the nearly round anti aligned shapes of retrograde orbits within the bar implies that they cannot be signi cantly populated in any self consistent model of a bar see also Teuben and Sanders 1985 However these orbits do play an important negative role in self consistent bars since the remarkably large regular regions that surround this family in Figure 13 and in most other studies mean that it reserves77 a substantial fraction of the phase space volume which the stochastic orbits cannot enter 40 Sellwood and Wilkinson Irregular orbits raise a major dif culty for exactly self consistent models that would survive inde nitely though for practical purposes this may be too abstract a theoretical requirement As their name implies irregular orbits follow chaotic trajectories with no periodicities whatever The density distribution produced by a population of stars having irregular orbits can never be steady therefore It is not clear what this means in practice bars in galaxies are typically quite young dynamically perhaps only 50 rotation periods We might speculate that a signi cant fraction of irregular orbits may not have begun to explore the full space available to them or the changes they cause could be suf ciently small and slow that the bar can adjust continuously without being weakened or destroyed It is far from clear how these ideas can be tested 49 Tour de force Pfenniger 1984b presents by far the most extensive attempt to show that some mathematically convenient elliptical bar model could be made self consistent Adopting Schwarzschild7s approach 35 but with a non negative least squares algorithm instead of linear programming he managed to obtain self consistent models which to our knowledge remain the only solutions in the literature for a rapidly rotating two dimensional bar In constructing his orbit library Pfenniger considered that the density distribution had converged to a steady state when the largest change in the occupation number of any cell dropped below 05 upon doubling the integration time Most regular orbits converged after a short integration but when his criterion proved impractical for some irregular orbits he imposed an arbitrary maximum integration time of about 550 bar rotation periods Pfenniger was able to use his approximately self consistent solutions to calculate velocity and velocity dispersion elds and found at least four different forms The simplest ow for the maximum angular momentum consisted of directly rotating elongated owlines within the bar and circular owlines corresponding to the inner ring This ow pattern was the one which agreed most closely with observations of early type galaxies and N body models Other more complicated eddying ows were also possible Some retrograde orbits were always required involving typically between 10 and 30 of the mass and populating the lens like part of his assumed elliptical bar model Models could be constructed without irregular orbits but usually about 10 of irregular orbits were required The dispersion elds showed little anisotropy with the dispersion decreasing from the central value by a factor of about two by the co rotation radius 492 N body model An entirely different approach was adopted by Sparke and Sellwood 1987 who examined the orbital structure of a bar formed in an N body simulation They were able to nd many ofthe usual orbit families in the frozen potential of the model and determined from their distribution in the SOS that the majority of particles making up the bar were either trapped or semi trapped about the 1 family Their N body bar had a quite realistically rectangular appearance yet they found somewhat surprisingly that the 41 family was of little importance Instead the orbits librating about the 1 family seemed to be responsible for the rectangular shape Sparke and Sellwood noted that their two dimensional bar model was remarkably robust and could adjust essentially immediately to major alterations of the global potential Unfortunately their conclusion applies only to models restricted to two dimensions once particles are allowed to move normal to the symmetry plane the bar appears to suffer another type of instability see 101 Barred galam39es 41 493 Photometric models There is a recent welcome trend in the literature to adopt mass distributions which bear some resemblance to the light distributions of barred galaxies which nature has constructed self consistently One of the rst such attempts was made by Kent and Glaudell 1989 for the well studied SBO galaxy NGC 936 This galaxy is one of the most favourable for such a study Figure 2 being bright and nearby largely free from star formation regions and obscuring dust and viewed from an ideal angle Unfortunately even for this galaxy major free parameters remain essentially undetermined by the observational data the most important being the pattern speed of the bar and the mass to light ratio of the stellar populations the latter is most unlikely to be a universal constant throughout one galaxy Kent and Glaudell attempted to constrain the mass to light ratio from the observed velocity eld and experimented with two pattern speeds Their results were rather weakly in favour of the higher of the two pattern speeds which places co rotation a little beyond the end of the bar A more recent study of the same galaxy based on new photometric data has been under taken by Wozniak and Athanassoula 1992 Instead of trying to decompose the model into components they adopt a mass model based directly on the observed light distribution They concur with Sparke and Sellwood 1987 that the rectangular bar shape is supported by orbits trapped about 1 and owes little to the 41 family 5 Threedimensional bar models The N body experiments of Combes and Sanders 1981 provided the rst indication that an exclusively two dimensional treatment of barred galaxies is inadequate although their result was not understood at the time We now believe that the thin disc approximation cannot be invoked for bars for two distinct reasons rstly vertical resonances occur in the bar which couple horizontal to vertical motions and secondly thin bars are subject to out of plane buckling instabilities We describe buckling modes together with other forms of bar evolution in 10 after discussing bar formation Thus a simple addition of small vertical oscillations to the orbits discussed in the previous section would give a seriously incomplete description of the three dimensional dynamics of bars Work on fully three dimensional models of rapidly rotating bars is still in its infancy however 5 Vertical resonances For a nearly circular orbit in a weakly perturbed potential we expect resonances between the vertical oscillation and a rotating m fold symmetric perturbation wherever leCU 7 9p nszr here HZ is the frequency of oscillation normal to the symmetry plane and n is an integer The vertical co rotation n 0 resonance is of no dynamical importance but the n 31 0 resonances could couple motion in the plane to vertical excursions Binney 1981 The n i1 resonances are sometimes known as vertical Lindblad resonances77 and the n i2 resonances give rise to the much discussed Binney instability strips77 which occur only for retrograde orbits in the inner galaxy or far outside co rotation for direct orbits 711 mn in our notation n in Binney7s It is useful to compare HZ to H We write Poisson7s equation for an amisymmetm39e mass distribution as 1 6 6lt1gt 62 i 4 G 28 r 37quot T 37quot 322 7 W p 42 Sellwood arid Wilkinson Recognizing that the combination r is the square of the circular velocity the rst term vanishes in the mid plane of a galaxy having a at rotation curve The remainder of the equation then gives us the standard result HZ l47TGPO 29 where p0 is the density in the mid plane To estimate a we note that H2 293 exact for a at rotation curve and that Q 4Gp where p is the mean spherically distributed density of matter interior to the point in question Thus the ratio KZ 4 p0 which is large wherever the self gravity of the disc is important Tremaine 1989 The inequality KZ gt H holds for the majority of stars which remain close to the plane and implies that the n i1 resonances lie further from co rotation than do the horizontal Lindblad resonances which delimit the spiral pattern The perturbing potential for spiral waves in a cool disc can therefore be expected to be negligible at the rst and all subsequent vertical resonances Thus it is legitimate to ignore vertical resonant coupling when the orbits of stars are nearly circular the disc is thin and the non axisymmetric component of the potential weak Obviously expression 30 for KZ fails in a strong bar Not only does a strong bar add a large non axisymmetric term to 29 but stars also move on highly eccentric orbits Therefore local estimates of the vertical oscillation frequency cease to be meaningful for an orbit and the existence of resonances can be determined only from orbit integrations Pfenniger 1984a was the rst to show that vertical resonances were important within the bar of a reasonably realistic three dimensional model 52 Periodic orbits in three dimensions Unfortunately very few studies of three dimensional periodic orbits have been published for models which bear much resemblance to barred galaxies the large majority are concerned with slowly rotating tri axial ellipsoids We mention this work insofar as it seems relevant but space considerations preclude a thorough review of the literature on the orbital structure of slowly rotating ellipsoidal models see 69 de Zeeuw and Franx 1991 521 Orbital stability As for rotating two dimensional potentials Jacobi7s integral is con served for all orbits 432 in a potential which is steady in rotating axes There are therefore four adjustable coordinates for the shooting method 44 which are usually yyz2 and again we seek solutions such that TWO 0 For consistency with the previous section we use y for the bar minor axis The stability test for a periodic orbit now requires the determination of the eigenvalues of a 4 gtlt 4 Jacobian matrix of partial derivatives equation 28 As for two dimensions the orbit is stable when all eigenvalues lie on the unit circle However there are six physically distinct types of instability which have been discussed for a very similar celestial mechanics problem by Broucke 1969 Hadjidemetriou 1975 and Henon 1976 and for bars by Pfenniger 1984a and Contopoulos and Magnenat 1985 The phenomenon of samplers instability which appears only in systems with more than two degrees of freedom has been discussed extensively in this case all four eigenvalues are complex and not located on the unit circle What causes an orbit sequence to become complex unstable is not well understood but the transition frequently occurs at boundaries of stochastic regions Magnenat 1982b Barred galtwies 43 522 Notation A very substantial fraction of the work in this area has been carried out by the group at the Geneva Observatory Notwithstanding the importance of their work7 we nd the notation they adopt7 which is an attempt to generalize from two dimensions7 so cumbersome that we feel unable to use it here To our knowledge7 there are at least four other competing systems in the literature 7 an indication perhaps of the new and unsettled nature of the eld7 and a source of confusion to all but the expert Since no consensus on nomenclature has been reached7 and we do not nd any suggested systems attractive7 we have developed our own It does7 however7 share features of some of the other systems We extend the ml notation in the plane to become mi il7 where m is the number of radial oscillations7 when viewed from the rotation axis as for two dimensions and n the number of vertical oscillations before the orbit closes after l rotations about the centre Unfortunately7 this nomenclature needs to be supplemented to indicate whether the orbit is symmetric or anti symmetric about the Lz plane7 ie the plane through the bar centre normal to its major axis When there is a need to distinguish these we add a subscript thus7 mnsl and mnal refer to the symmetric and anti symmetric families respectively We illustrate this nomenclature with the orbits discussed in this section Our system7 like most others7 is too rigid to describe transition regions between families A glance at Figure 12 shows that distinctly named families frequently join smoothly into a single orbit sequence in two dimensions any notation system based upon orbital shapes will also ascribe different names to parts of a sequence in three dimensions The gradual change in the properties of orbits over the transition region is not allowed for in notation systems which require an abrupt shift to a new designation at some arbitrary point along a sequence We regard this weakness as more than outweighed by the advantage of a notation which clearly indicates the shape of the orbit away from transition regions 523 Periodic orbit families Pfenniger 1984a investigated orbits in the combined poten tial of a rotating7 tri axial Ferrers bar with axes in the ratio 14107 and a thickened Kuz7min disc Miyamoto and Nagai 1975 He made the bar rotate about the short axis at a rate which placed co rotation at the end of the long axis First he examined the periodic orbits in the equatorial plane and found that the main 1 family had something like the usual form As his model was insuf ciently centrally condensed to contain ILRs when axially symmetrized7 it could not support the perpendicular families 2 and 3 of periodic orbits in the plane The 1 family was vertically stable throughout most of the inner part of the bar7 but was vertically unstable over several short stretches at higher energies These instability strips lay between pairs of bifurcations from which new periodic orbit families with non zero vertical excursions branched off The rst pair of bifurcations gave rise to families of orbits which retained an oval appearance from above7 while developing small wrinkles when viewed from the side Figure 14 top These were symmetric and anti symmetric 241 families As both formed short sequences in the characteristic diagram and soon rejoined the main 1 family7 they were probably of little importance At higher energies7 the next two vertical families to appear were 441 orbits again symmetric and anti symmetric7 Figure 14 middle having much more extensive characteristic curves They had the distinctive shape of the 41 orbits in the 12 This differs from the super cially similar notation used by Schwarzschild7 which counts oscillations parallel to the coordinate axes His notation is well suited to nonrotating bars 44 Sellwood and Wilkinson F a 39 h 2491 224321 l Ckfoka Figure 14 Six three dimensional periodic orbits found by Pfenniger 1984a the three on the left are symmetric and those on the right are their anti symmetric counterparts We have re labelled these orbits using our notation plane as well as a 41 oscillation normal to the plane A third pair of bifurcations led to the 461 orbit families Figure 14 bottom which Pfenniger again viewed as less important He found that all three anti symmetric families were generally unstable over most of their length but the symmetric sequences were stable over long regions Only the 441 families remained when he reduced the bar mass substantially It seems likely that the low central density of his model prevented him from nding any m21 families Pfenniger also tested the effect of changing the axial ratios of the bar He found that both the horizontal and vertical stability of the 5131 family of orbits decreased as the bar became thinner in the plane of the disc when the axis ratio in the plane exceeded N 5 7 most of the 511 sequence became unstable Increasing the bar thickness perpendicular to the plane as far as making it prolate also improved both the vertical and horizontal stability probably because the non axisymmetric component of the potential weakens as the bar is made thicker Most subsequent studies have been concerned with slowly rotating tri axial ellipsoids though some of the results are relevant to barred galaxies In particular directly rotating 221 resonant families seem to be most relevant to galactic bars The 222821 orbits are com monly known as bananas and the Geneva group uses the deplorable term anti bananas to describe the 2251 orbits13 Both these families bifurcate from the main 5131 family in most mod 13 It should be noted that these orbits in a rapidly rotating bar differ from those also termed anti bananas in non rotating bars Miralda Escud and Schwarzschild 1989 which pass through the exact centre of the bar Barred galam39es 45 els Mulder and Hooimeyer 19847 Cleary 1989 but they can also be found as bifurcations from 2 when the bar rotates suf ciently slowly Udry 1991 The family in the plane seems always to be vertically unstable between the two bifurcation points7 which are generally quite close to gether Most authors nd that the symmetric family is stable and the anti symmetric unstable from their bifurcation points7 but that stability is exchanged between them at a higher energy Mulder and Hooimeyer 1984 found an additional intermediate family connecting the two at the energy where stability was exchanged7 but Udry 1991 could not7 unless the potential was perturbed Udry also maps out the limiting bar axis ratios for which the 221 families can be found in a model having a tri axial modi ed Hubble density pro le He nds that the short axis should not be more than 30 to 40 of the long axis7 with only a slight variation over the entire range possible for the intermediate axis He also notes that these values are hardly affected by rotation7 although he did not investigate very rapidly rotating models The addition of a small Plummer sphere at the centre of the mass model further con ned the existence of these families to much atter bars7 however 53 Structure of a three dimensional N body model Pfenniger amp P riedli 1991 have carried out a detailed study of the orbital structure of one of their three dimensional N body simulations As the bar in their model thickened in the z direction quite markedly during its rst few tumbling periods see 1017 they chose two different moments for their study one soon after the bar formed as a thin structure7 and the second much later when evolution seemed to have ceased They rst searched for periodic orbits in the frozen potentials at the two instants7 both in the raw potential of the model and when they imposed re ection symmetry about the three principal planes The pattern speed of the bar dropped by some 20 and the approximate axis ratios of the bar shape rose from N 1 042 033 to 1 051 040 between the two moments analysed In the symmetrized cases7 they found that the planar 1 family was stable7 both horizontally and vertically7 from the centre almost to the 31 bifurcation7 except for a short vertical instability strip The perpendicular families in the plane 2 and x3 were absent from their model7 but only marginally so The bifurcations at either end of the short vertical instability strip in the main 1 family lead to the symmetric and anti symmetric 221 families7 illustrated in Figure 15 As usual7 the anti symmetric family was unstable and the symmetric family stable near the plane7 but they exchanged stability at a higher energy The vertical extent of the stable part of the symmetric family had decreased quite markedly7 and the bifurcation points moved out along the bar7 by the later time There were no simple periodic orbits which remained precisely in the plane in the raw un symmetrized potential of the N body model7 because the potential was not exactly symmetric in 2 at either time Nevertheless7 they could trace families generally resembling those in the symmetrized model7 but with two important differences rst7 the vertical bifurcations7 which were of the pitchfork type in the symmetrized model7 became resonant gaps in the raw poten tial7 and second7 the 22a1 families became much more stable while the 22s1 families lost still more of their stability These stability di ferences were particularly marked at the later time Pfenniger and Friedli noted that the majority of particles in the bar seemed to follow quasi periodic orbits trapped about the 1 and the 221 families Very few orbits were retrograde or belonged to any of the other families they found from their periodic orbit analysis For energies approaching the Lagrange point7 they found larger and larger fractions of irregular orbits 46 Sellwood and Wilkinson ITIIFIYIXIIIFrII lllll lll39lllllllll ITITV IITIIIIIIIIIT J ll39llllllllillllll ll39llflllllil39lli Jillilllllllillll TillllllllllllillllllllllillllllilLLlJ 4 2 O 2 4 4 2 O 2 4 X Y Figure 15 Three orthogonal projections and a tube view showing the three dimensional shapes of both the symmetric and anti symmetric 221 orbits The symmetric orbit is drawn as a continuous line and the anti symmetric is dashed the dotted lines show an intermediate case Reproduced from Pfenniger and Friedli 1991 Given the stability properties of the 22221 families it seems likely that stars in the box shaped nal model are trapped about the 22azl family a conclusion supported by the pre liminary orbit analysis of his own models performed by Raha private communication These anti symmetric orbits were rst reported by Miller and Smith 1979 who found them in large numbers in their slowly rotating ellipsoidal model Pfenniger and Friedli note that their model also supported vertical bifurcations along the retrograde 514 family which gave rise to 2121 orbits also known as anomalous retrograde orbits Their study of the orbital make up of the bar makes it clear that these are of little importance in fast bars though they have been of major interest for slowly rotating tri aXial systems Heisler et al 1982 Magnenat 1982a Mulder and Hooimeyer 1984 Martinet and Pfenniger 1987 Martinet and de Zeeuw 1988 Cleary 1989 Martinet and Udry 1990 etc 54 Stochasticity m threedimensions The most important difference between two and three degrees of freedom is that the phe nomenon of Arnold di usion appears Regular orbits form boundaries which irregular orbits cannot cross for test particles in two dimensions these invariant tori divide phase space into separate volumes so that an irregular orbit can be semi trapped inside an invariant curve The extra degree of freedom in three dimensions means the chaotic regions are no longer iso lated from each other by the invariant surfaces which have too few dimensions and are thought to be connected into a single network known as the Arnol d web Self consistent models containing a signi cant fraction of chaotic orbits would be dif cult to construct since any chaotic orbit lls a volume bounded only by its energy surface which is always more nearly spherical than the density distribution giving rise to the potential Models containing some stochastic orbits are therefore likely to be only quasi stationary Since bars Barred galam39es 47 have existed for some 50 orbital periods only the fact that some stochastic orbits will cause them to evolve on a longer timescale may not be of great concern as the essentially stationary three dimensional models of Pfenniger and Friedli 1991 show Most orbits in non rotating three dimensional ellipsoids are of the box or short and long axis tube types Schwarzschild 1979 Binney and Tremaine 1987 which are regular orbits trapped about the simplest periodic orbit families However the introduction of even slow rotation appears to change the orbital structure drastically 7 many orbits which were regular in stationary bars become stochastic There seem to be a number of additional ways apart from slow rotation to foment stochasticity but only rapid rotation is known to reduce it 47 Martinet and Pfenniger 1987 investigated the effect of a mass concentration in the galaxy core and showed that for even a small mass the z motion close to the centre was quickly desta bilized Hasan and Norman 1990 con rmed that a central mass concentration is extremely effective at causing orbits close to the mass to become chaotic particularly once the central condensation contains more than 5 of the total mass Udry and Pfenniger 1988 found that stochasticity rises when the bar is strengthened 69 by making it narrower or squaring off its ends and again when the central concentration was raised They also examined the effects of graininess in the potential a very reasonable degree of granularity destabilized yet more regular orbits Though many questions remain unanswered collectively these results suggest that weak rapidly rotating bar models should have the fewest chaotic orbits and strong bars which end well before co rotation would have the most 6 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 ow 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 ow pattern and the calculated gas behaviour 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 rst proposed by Prendergast unpublished c1962 Finally the speci c angular momentum of gas elements changes with time causing signif icant radial ows 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 ow stops Almost all the theoretical work and simulations have neglected motion in the third dimen sion This approximation may still be adequate notwithstanding the existence of transient bending instabilities 101 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 ow of gas is accelerated as the material passes through vertically unstable regions 48 Sellwood and Wilkinson 6 Observations of gas in barred galasies Most data come from optical or 21cm Hl observations and much less is known ofthe 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 61 Gas distribution The distribution of neutral hydrogen within each galaxy shows considerable variation Neutral hydrogen appears to be de cient within the bar in a number of galaxies 69 NGC 1365 Ondrechen and van der Hulst 1989 and NGC 3992 Gottesman et al 1984 On the other hand counter examples with signi cant H1 in the bar are NGC 5383 Sancisi et al 1979 NGC 3359 Ball 1986 NGC 4731 Gottesman et al 1984 NGC 1073 England at al 1990 and NGC 1097 Ondrechen et al 1989 The CO is sometimes distributed in a ring around the bar eg Planesas et al 1991 and sometimes concentrated towards the nucleus 69 Sandqvist et al 1988 Early type barred galaxies contain little gas in common with their unbarred counterparts eg 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 suf cient Hl to be mapped found that in both cases the gas was concentrated in an outer ring 612 Kinematics The position velocity maps of gas in a barred galaxy indicate that the ow pattern is more complicated than the simple circular streaming sometimes seen in an approximately axisymmetric galaxy In general systematic variations in the observed velocity eld 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 eld 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 H1 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 eg 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 eg 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 Hl data from the Very Large Array VLA has become available in recent years and the number of barred galaxies with well determined velocity elds is rising albeit slowly We mention individual papers in 67 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 Barred galaaies 49 19917 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 two early type galaxies and such cases are believed to be rare 613 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 straight7 lying at an angle to the bar as in NGC 13007 or curved as in NGC 6782 and NGC 1433 Sometimes predominantly straight lanes curve around the centre to form a circum nuclear ring Dust can also be distributed in arcs and patches across the bar NGC 1365 Figure 1 is a good example 614 Evidence for shocks Direct observations of steep velocity gradients across dust lanes7 which would be the most compelling reason to believe these are shocks7 have been hard to obtain The two best examples are for NGC 6221 Pence and Blackman 1984a and for NGC 1365 Lindblad and Jo39rs39ater 1987 Evidence for gas compression also comes from the distribution of molecular gas7 through CO emission7 which appears to be concentrated in dust lanes eg Handa et al 19907 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 eg Ondrechen 19857 Hummel et al 1987a7 Tilanus 1990 7 the enhanced emission is consistent with gas compression7 but could also have other causes 615 Star formation and other activity It has been noted frequently eg Tubbs 19827 and references therein that the distribution of young stars and Hll regions is not uniform in barred galaxies Stars appear to be forming proli cally near the centres Hawarden et al 19867 Sandqvist et al 19887 Hummel et al 1990 and at the ends of the bar7 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 bar7 eg NGC 4321 Arsenault et al 19887 Arsenault 1989 Dense concentrations of molecular gas are also sometimes found near the centres of barred galaxies eg Gerin et al 1988 It has also been noted by several authors eg Simkin et al 19807 Arsenault 1989 that active galactic nuclei are somewhat more likely to occur in galaxies having bars7 than in those without 62 Modelling the ISM When speaking of shocks etc in a gas ow7 it is customary to think of a continuous uid having a well de ned sound speed Unfortunately7 the ISM inter stellar medium is not that simple7 which prompted Prendergast 1962 to muse that it is unclear what to assume for the equation of state It might seem natural to assume that the inter stellar gas in the neighbourhood of the Sun7 which is the best studied portion of the ISM7 has properties typical of that throughout all galaxies Here7 the bulk of the gas mass is contained in cool7 dense clouds which orbit ballistically7 virtually unaffected by external pressure forces7 except when in collision with another dense cloud By contrast7 the bulk of the volume is lled with high temperature gas at a much lower density7 which is in rough pressure balance with the dense material 7 the so called hot phase The balance between the components is regulated by star formation and supernovae for more information7 see reviews by Cox and Reynolds 1987 or Spitzer 1990 50 Sellwood and Wilkinson Unfortunately the star formation rate and probably also the supernova rate7 though no data are available within bars seems to differ from that near the Sun7 and the ISM in bars may have somewhat different properties This variously damped and heated multi phase stew Toomre 1977 is believed to experi ence shocks of some form or other in the intriguingly located dust lanes We cannot realistically hope to understand the large scale behaviour of the ISM if we include the intricate small scale dynamics of each uid element7 and most calculations assume some gross physical properties for the medium The pervasive hot phase is the only component which can reasonably be described as a smooth uid on galactic scales7 but both the sound speed and the Alfven speed are likely to be well in excess of 100 km sec l7 implying that there is little chance of it being shocked by a potential perturbation with relative motion a fraction of the orbital velocity One approach has been to treat the dense material as a collection of ballistic particles having a nite cross section for collision Miller et al 19707 Schwarz 19797 Matsuda and Isaka 19807 Combes and Gerin 1985 There is some disagreement over whether to dissipate all or just some fraction of the energy in the collision7 whether to merge the colliding particles and what to assume for the collision cross section It is not unreasonable7 however7 to view the collection of cool clouds as a uid with a sound speed of the order of the velocity dispersion of the clouds7 which is typically 5 7 10 km sec l The conditions under which this might be valid were examined by Cowie 19807 who attempted to calculate an equation of state for the cloud ensemble This vastly simplifying assumption has led to a rival group of papers which calculates the gas ow as a continuous uid using conventional two dimensional uid dynamical codes A number of different codes have been tried van Albada et al 1982 assessed the relative performance of many common techniques and a multi grid method was later introduced by Mulder 1986 Some authors eg Roberts et al 19797 with an understandable desire for yet higher spatial resolution7 advocate 1 D codes which neglect pressure forces normal to the ow lines Some techniques are of intermediate type7 such as the beam scheme Sanders and Prender gast 19747 which has proved very popular7 and smooth particle hydrodynamics or SPH Lucy 19777 Gingold and Monaghan 1977 Both codes combine aspects of the previous two distinct approaches Hernquist and Katz 1989 describe a three dimensional uid dynamical scheme which uses SPH with self gravity As the gas in all three types of code obeys equations which are at best very crude approximations to the real dynamics of the ISM7 a discussion of which is intrinsically the best misses the point Several authors argue the virtue of low numerical viscosity in high quality uid codes7 without pausing to consider the extent to which the ISM differs from an inviscid gas with a simple equation of state In fact7 bulk viscosity may be the most important physical property distinguishing the gas from the stars eg Sanders 1977 Numerical viscosity is7 of course7 undesirable because its properties and magnitude are set by the nature of the numerical code7 grid cell size7 collision cross section etc7 whereas it would be preferable to employ a viscous coef cient related to the properties of the medium In summary7 it is very helpful to have tried a variety of codes having different numerical weaknesses7 because we gain con dence in results which all can reproduce7 and learn to be suspicious of those unique to one type of code Barrcd galacics 51 63 Strcamlincs and pcriodic orbits Because the velocity dispersion of the gas clouds is so much lower than their orbital speeds the in uence of pressure77 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 ic the gas must have a unique stream velocity at each point in the ow 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 as in 42 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 crowding77 is greatest Each spiral arm winds through at most 90 per resonance crossed As Sanders and Hunt ley7s rst 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 sub sequent paper Huntley ct 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 ow may then remain aligned with the bar all the way from the centre to co rotation cg 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 64 Strong bars Streamlines still try to follow periodic orbits even in strongly non axisymmetric potentials though it becomes increasingly dif cult to nd circumstances in which the orbits can remain nested not only can adjacent orbits cross but a periodic orbit can also cross itself cg Fig ure 11 Because this greatly complicates the relationship between periodic orbits and stream lines we nd 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 cg Landau and Lifshitz 1987 108 we can think of the gas ow within a bar as a layer of shallow water circulating in a rotating non axisymmetric vessel having the shape of the effective potential cg Figure 10 for a rapidly rotating bar this has the shape of a non circular volcano crater see 432 As the crater rim de nes co rotation the water within the crater ows in the same sense as the bar It ows along the sides of the crater but has too much momentum to be de ected round the end and back along the far side by the comparatively weak potential 52 Sellwood and Wilkinson Figure 16 The gas ow pattern in one of Athanassoula7s 1992b simulations in which the inho mogeneous bar is positioned across the diagonal of each frame a shows a grey scale representation of the gas density distribution highest densities are white b shows the velocity vectors and a few streamlines in the restframe of the bar c shows some periodic orbits from the 1 family on an expanded scale and d shows the loci of density maxima and the outline of the bar on the same scale as in a and gradients Instead it rushes on past the major axis of the potential and on up the sides of the vessel nally turning back when the ow stalls The hydraulic jump which must form where fresh material encounters the stalled ow 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 rst revealed in the uid dynamical simulations by Sorensen et al 1976 and have been reproduced many times Roberts et al 1979 Sanders and Tubbs 1980 Schempp 1982 Hunter et al 1988 etc Figure 16 shows such a result from the high quality simulations by Athanassoula 1992b As expected she nds that shocks develop only when the 951 family of orbits possess loops or are at least very sharply curved 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 11 and 13 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 insu iciently centrally concentrated then the shocks lie close to the bar major ax1s Barred galam39es 53 65 Driven spiral arms The spirals that Sanders and Huntley 1976 and Schwarz 1981 were able to produce extended out to and even a little beyond the OLR However many authors have reported that they are unable to reproduce such extensive spiral responses in passive gas to forcing by an ellipsoidal bar model instead of an oval distortion to the potential This is because the quadrupole eld of a realistic bar which ends near co rotation falls off too rapidly at larger radii to induce a spiral response in the gas Figure 9 In order to maintain a spiral response well beyond the bar Roberts et al 1979 made the rather ad hoc assumption that the bar near the centre goes over to a trailing spiral perturbation beyond co rotation the whole pattern rotating at the same rate Hunter et al 1988 added a co rotating oval distortion for similar reasons Spirals are more likely to be independent patterns as in normal galaxies and probably have a different pattern speed Sellwood and Sparke 1988 Such a suggestion would seem to imply a random distribution of phase differences between the bar and the start of the arms which it is generally felt conflicts with the observed situation Sellwood and Sparke point out that this is not a valid objection for two reasons Firstly a phase difference can in fact be seen in a number of barred galaxies 7 even where a two arm grand design77 spiral pattern dominates 69 NGC 5383 Figure 2 Secondly contour plots of the non axisymmetric density in their model show that the spiral arms appear to the eye to be joined to ends of the bar for most of the beat period A review of the problems and ideas of spiral arm generation in disc galaxies would take us too far from the subject of barred galaxies The most convincing evidence for two separate pattern speeds in a galaxy is that for NGC 1365 Figure 1 Ironically the impressive grand design spiral arms of this galaxy have frequently been cited as a clear example of bar forcing yet the kinematic data presented in 67 strongly support two separate co rotation radii for the bar and spiral patterns Weaker evidence can also be found for other galaxies 66 Angular momentum changes Whenever gas is distributed asymmetrically about the major axis of the potential it will expe rience 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 swept up 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 suf ciently high to have no ILRs but in his thesis Schwarz 1979 also reports inner ring formation in a different model having an ILR Schwarz nds that the high rate at which gas is swept up into rings depends only weakly upon his numerical parameters the collision box size coef cient of restitution etc Since the torque responsible for these radial ows 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 ow rate Schwarz7s ow rates are 54 Sellwood and Wilkinson 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 of al 1980 proposed a causal link between the in ow 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 in ow 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 67 Comparison with observations NGC 5383 Figure 2 is probably the most extensively studied and modelled barred galaxy Sanders and Tubbs 1980 made a systematic attempt to model the gas ow pattern measured by Peterson of al 1978 and Sancisi of al 1979 By varying the bar mass axis ratio pattern speed and other parameters they were able to nd a model which broadly succeeded in reproducing the qualitative features of the observed ow 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 ow pattern within the bar in more detail They ran simulations to determine the ow pattern when co rotation was close to the bar end and experimented mainly with a range of mass to light ratios for the bar again their best model resembled the observed ow pattern within the bar though still not impressively so It is possible that their low resolution beam scheme code precluded a better t Pence and Blackman 1984b found that the velocity eld 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 at al 1990 is too nearly face on for the kinematic data to constrain a model In both the other two NGC 3992 Hunter of al 1988 and NGC 1300 England 1989 they encountered considerable dif culties in modelling the outer spiral When a pure bar model failed to produce a suf ciently 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 but are independent dynamical structures of the type described in 65 NGC 1365 Figure 1 has also been observed extensively though no good model for the whole galaxy has yet been published Teuben of al 1986 nd 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 2 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 7 a reassuring circumstance which is also corroborated by the presence of the offset dust lanes along the strong bar see 64 However Ondrechen and van der Hulst 1989 note that the inward direction of the gas ow 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 Barred galaries 55 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 dif culty in assigning a pattern speed for NGC 7723 the dust lanes in NGC 1365 Figure 1 and NGC 1300 Sandage 1961 cross the spiral another7 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 19877 yet this is close to the bar end this may be an example where the spirals rotate more rapidly than the bar 7 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 ow pattern provides useful constraints on the properties of the bar ii The gas ow 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 speed7 but are offset only if the 2 family is present iv The outer spiral arms usually cannot be modelled without assurning some additional non axisyrnrnetric component to the potential v Not one of the well studied cases provides evidence that the spiral patterns are driven by the bar7 while there is frequently a suggestion that the spiral arms have a lower pattern speed than does the bar 7 Rings and lenses Many galaxies7 both barred and unbarred7 exhibit rings which are believed to lie in the disc plane These are thought to have an entirely different origin and properties from the much rarer polar rings7 so named because they lie in a plane alrnost perpendicular to that of the disc We do not discuss polar rings in this review see eg Whitrnore et al 1990 Unlike the spiral patterns in barred galaxies just discussed7 there is considerable evidence that many rings share the same pattern speed with the bar7 and therefore seem very likely to be driven responses to forcing by the bar 7 Observed properties of rings Statistical properties of some 1200 ringed galaxies selected from the southern sky survey are presented by Buta 1986a these appear to be quite representative of the complete catalogue Buta 19917 which will contain about twice this number Three major ring types are distin guished by the radii7 relative to the bar rnajor axis7 at which they occur 71 Outer rings Outer rings are the largest7 relative to the host galaxy7 having a diameter some 22 i 04 times the bar major axis Korrnendy 1979 A good example is NGC 2217 in Figure 2 and others include NGC 12917 NGC 2859 de Vaucouleurs 1975 and NGC 3945 Korrnendy 1981 The frequency of outer rings is dif cult to estimate because they could be missed on all but the deepest plates early estirnates de Vaucouleurs 19757 Korrnendy 1979 suggested they occur in only 4 7 5 of all galaxies7 but Buta7s survey may well indicate a higher fraction Buta7 private communication 56 Sellwood and Wilkinson The ring is centred on the nuclear bulge and is thought to lie in the disc plane Statistical arguments indicate that outer rings have intrinsic axis ratios in the range 07 to 10 Athanas soula et al 19827 Schwarz 1984b7 Buta 1986a Buta notes that although the longer axis is usually perpendicular to the bar7 there is a signi cant sub population for which the ring is parallel to the bar True outer rings are not always easily distinguished from pseudo outer rings7 which occur when the outer spiral arms almost close 712 Inner rings lnner rings are somewhat smaller7 and generally have a diameter similar to the bar major axis 7 good examples are NGC 1433 and NGC 2523 in Figure 2 7 but are sometimes noticeably larger7 eg NGC 936 also in Figure 2 They are more common than outer rings7 but are found mainly in later types Kormendy 1979 reports that 76 of SBab SBc galaxies have inner rings7 while few early type galaxies have them lnner rings in barred galaxies are generally more elliptical than outer rings7 having an axis ratio in the range 06 7 095 Buta 1986a and are elongated parallel to the bar Schwarz 1984b with few exceptions14 Buta 1988 nds some evidence for non circular motion in the rings of a few nearby galaxies7 thereby con rming their intrinsic non circular shape Buta 1991 notes that some are more rectangular or even hexagonal the best example of hexagonal isophotes is for the weakly barred galaxy NGC 7020 Buta 1990b In galaxies having both types of ring7 the ratio of the outer to inner ring major axis diameters is on average 221 i 002 with a long tail to higher values Buta 1986a Both inner and outer rings tend to be bluer than the surrounding disc and many have Hll regions7 as do spiral arms Buta 19887 Buta and Crocker 1991 713 Nuclear rings A third type of ring has so far been found in relatively few systems7 eg NGC 1512 Jo39rs39ater 19797 NGC 1365 Teuben et al 19867 NGC 1097 Hummel et al 1987b7 Gerin et al 19887 NGC 4321 Arsenault et al 19887 NGC 5728 Schommer et al 1988 and NGC 4314 Garcia Barreto et al 1991a They are usually very small7 radius a few hundred parsecs7 nearly round7 and not aligned with the bar Buta 1986a An exceptionally large nuclear ring is seen in ESO 565 11 Buta and Crocker 1991 As they are hard to nd7 because of their small size7 their apparent rarity could again simply be a selection effect In some cases7 co incident radio and optical emission from discrete sources lying in the ring is found7 together with signi cant quantities of molecular gas and dust eg Sandqvist et al 1988 Hawarden et al 1986 also note an excess of 25pm emission from barred galaxies7 which they interpret as being due to enhanced star formation in a nuclear ring However7 Garcia Barreto et al 1991b point out that similar phenomena can also occur without a detectable ring Several galaxies contain short nuclear bars within the nuclear ring We discussed such features in 24 72 Lenses and oval distortions While inner rings are rare in early type galaxies7 some 54 of SBO SBa galaxies Kormendy 1981 manifest a attened ellipsoidal structure known as a lens Clear examples are NGC 5101 14 Exceptions are NGC 43197 which is a tidally interacting system7 and NGC 6300 in which two tightly wrapped arms make a pseudoinner ring Buta 1987 Barr6d 9oloi6s 57 Sandage 1961 and NGC 3945 and NGC 4596 Sandage and Brucato 1979 This feature is similar in size to the inner rings discussed above Kormendy 1979 notes that the bar usually lls the lens in one frequently the longest dimension The axial ratio in the disc plane is typically 09 i 0057 i6 they are slightly rounder than inner rings Comparatively few late type galaxies SBb to SBm types are classi ed as having a lens7 but many have a so called oval disc or distortion Kormendy suggests that lenses and oval discs are distinct phenomena7 on the grounds that the kinematic properties appear to be different the rotation curves of oval discs are at 69 NGC 47367 Kormendy 19797 whereas the rotation curve of a lens seems to rise with radius Kormendy 1981 Clearly more data are required to establish his case 73 Formation of iiii9s By far the most popular theory is that rings form from radial ows of gas driven by the bar The gas dynamical simulations by Schwarz 1979 showed that material gathers in rings where the radial ow7 caused by the spiral response to the bar7 ends at a major resonance of the pattern In this picture7 the outer rings lie at the OLR for the bar while the inner rings occur either at co rotation or at the 41 resonance The nuclear rings are thought to lie at the ILR for the bar 69 Combes and Gerin 19857 Buta 1986b7 Schommer 6t ol 19887 and the inner bar may be populated by stars on the perpendicular orbit family 2 69 Teuben 6t al 1986 There are two pieces of evidence in favour of this interpretation Firstly7 making plausible assumptions about the shapes of rotation curves7 Athanassoula 6t al 1982 concluded that radii were consistent with the hypothesis that the outer rings lay at the OLR while the inner were located at the 41 resonance Secondly7 and more convincingly7 the ring orientations t extremely well with the results from the simulations and suggest that rings trace the major periodic orbits ln particular7 Schwarz 19797 1981 found two orientations for the outer ring7 depending upon the extent of the original gas disc both cases seem to occur abundantly in nature7 and there are even a few galaxies which seem to possess both simultaneously Buta 1986a7 1991 However7 the theory does not offer convincing reasons for the existence of rings in unbarred galaxies 7 less common7 but by no means rare Buta 1991 7 and the absence of rings in other barred galaxies The rst might be explained by arguing that the bars are unseen in the optical 1 or that they have dissolved after rst forming a ring 103 The second may indicate a nite lifetime for these features The theory loses more of its appeal if7 as we be have argued7 spiral patterns in barred galaxies rotate at a rate different from the bar The spirals themselves may also drive gas radially7 but the radii of rings formed by the spirals cannot be expected to correlate with the resonances of the bar15 A way to salvage the theory7 is to suppose that outer rings are formed at the same time as the bar The bar formation process causes a substantial re arrangement of angular momentum in the disc7 and frequently forms a transient ring in the stellar component near the OLR for the bar 927 a weak example is shown in Figure 17 This is a much more ef cient way to accumulate material at the OLR than through slow forcing by the weak quadrupole eld of a steady bar The lifetime of the gaseous ring7 which must be formed at the same time as the stellar ring7 is likely to be much greater because the random motions7 15 Unless the pattern speeds are related in some special way 69 Sellwood 1991 58 Sellwood and Wilkinson which cause the stellar ring to dissolve can be dissipated in the gas The comparatively small fraction of outer rings observed may indicate that these are still rather short lived features and therefore that the bar may have formed recently in galaxies where they are seen The theory that bar driven radial ows form rings may account for the outer and nuclear rings but it is less clear that it can account for inner rings It has been suggested eg Schwarz 1984a that inner rings contain material trapped about the stable Lagrange points on the bar minor axis or on higher resonant orbit families This last suggestion is an attractive one to account for the rectangular or even hexagonal shapes of some inner rings Buta 1990a b Buta 1988 and Buta and Crocker 1991 have started to acquire more data on a few good candidates but detailed photometric and optical and radio kinematic data are required for several more cases in order to provide a serious test of the bar driven ring theory 74 Formation of lerzses At present there is no good theoretical interpretation of lenses and oval distortions A number of rather speculative ideas have been discussed in the literature none of which we nd entirely convincing Kormendy 1979 speculated that a lens is formed by the dissolution of a bar since current ideas suggest that bars formed from instabilities in the disc this hypothesis does not seem to account for the substantially higher surface brightness of the lens Bosma 1983 proposed that the lens is an inner disc formed earlier than the faint outer disc and Athanassoula 1983 suggested that a lens results from a bar instability in a high velocity dispersion disc The latter idea su fers from the same aw as does Kormendy7s and requires a cool population of stars to be also present to form the narrow bar Teuben and Sanders 1985 weigh in with a similar suggestion that the lens is made up of chaotic orbits while Buta 1990a suggests that lenses are no more than aging inner rings 8 Asymmetries Unfortunately many barred galaxies are still more complicated in that they depart strongly from the perfect bi symmetry which has been implicit in all our theoretical discussion so far Mild asymmetries are seen in virtually all spiral galaxies but gross asymmetries occur much more frequently in late type galaxies They are not con ned to barred galaxies as emphasized by Baldwin et al 1980 8 Observed properties In some cases the disc and spiral pattern is simply much more extensive on one side of the galaxy An example of this type is NGC 4027 de Vaucouleurs et al 1968 in which the velocity eld of the emission line gas remains centred on the bar Pence et al 1988 even though the bar is far from the centre of the outer isophotes Other examples for which there is no kinematic data available are NGC 4618 and NGC 4625 Odewahn 1991 Another type of asymmetry is for the velocity eld to be noticeably asymmetric while the light distribution is only mildly so Examples are NGC 2525 which has a 50 km s 1 asymmetry between ends of bar Blackman and Pence 1982 NGC 3359 Duval and Monnet 1985 NGC 55 Hummel et al 1986 and NGC 7741 Duval et al 1991 Bettoni and Galletta 1988 noted a slight displacement in the centre of symmetry of the velocity pattern for 6 out of the 15 galaxies in their sample Barred galam39es 59 In other cases7 the bar is displaced from both the optical and kinematic centre Two of the most extreme examples are the LMC where the rotation centre is nowhere near the bar de Vaucouleurs and Freeman 1972 and NGC 1313 Figure 2 in which the rotation centre is at one end of the bar Marcelin and Athanassoula 1982 82 Models There have been very few attempts to model such asymmetries Marcelin and Athanassoula 1982 built a mass model based on photometric measurements of the galaxy NGC 13137 and solved for the gas ow assuming a uniform rotation of the eccentric mass distribution about the rotation centre They obtained quite a good match between their heuristic model and the observed velocity eld7 but declined to speculate as to why the mass distribution was so asymmetric or how it could remain so Colin and Athanassoula 1989 followed up this study with a similar treatment of other offset bar geometries There are no really convincing theories for the origin or persistence of these asymmetries It is probably signi cant that these late type galaxies have almost solid body rotation curves7 so it would take a very large number of orbital periods for asymmetric structures to be torn apart by the shear The rate at which asymmetric patterns are sheared can be further slowed if they are kinematic density waves and not material features Baldwin et al 1980 It seems likely that they originate through interactions with other galaxies or accretion of dwarf companions7 but there is also a possibility that they arise from m 1 type instabilities No detailed work appears to have been done to develop either hypothesis A further possibility is that they are primordial7 and that such galaxies are simply dynamically very young 9 Origin of bars The rst N body simulations of collisionless stellar discs Miller and Prendergast 19687 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 Figure 17 shows a recent high quality simulation illustrating this behaviour While this instability offers a natural explanation for the existence of bars in some galaxies7 it has proved surprisingly dif cult 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 galaxies7 it is hard to overstate its importance for disc galaxy dynamics The bar instability has therefore been repeatedly re examined from a number of directions7 which have all tended to con rm that rotationally supported discs suffer from vigorous7 global7 bi symmetric instabilities It would take us too far from the subject of this review to discuss the bar instability in great depth7 and we con ne 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 91 Global analysis A full stability analysis which leads to an eigenvalue problem for normal modes of an axisym metric stellar disc was rst formulated by Shu 1970 and by Kalnajs 1971 Though much can be formally deduced from this approach Kalnajs 19717 19777 it has proved very dif cult 6O Scllwood and Wilkinson d 0quot o u Q I 1 Jquot Q 39o I l 39 quot I 39 039 t c A Q I 43 23 quot 39 uquot 39 quot39 quotWm391 3 g6 39 39v39quot quot R o O 4 quotg39 w 39 I39i 39 u r quot3 39 o u o n 1 3 a I I 39 I 4191 39 i I t 39 3 539 r 31 039 cl 39 I Figure 17 The formation of a bar through dynamical instability in a largely rotationally supported disc The model is a completely self gravitating two dimensional isochrone6 model whose dominant unstable mode was calculated by Kalnajs 1978 100K particles are employed but only 5K are shown in each frame the times are marked in units of xCLBGM and the circle is drawn at a radius of 6180 where a is the isochrone scale length and M is the mass of the untruncated disc The linear growth of the mode can be detected by Fourier analysis before time 150 and the estimated growth rate is some 16 below the predicted value The discrepancy is probably caused by the gravity softening scale 005a which was introduced in the simulation Barred galam39es 61 to nd eigenvalues in practice We describe Kalnajs7s method and its practical dif culties of implementation here not only because it gives a vivid illustration of the analytical problems encountered in galactic dynamics but also because it reveals the power and limitations of action angle variables The limited results that have emerged from this work are however of immense value as they provide predictions against which we can test the N body results Moreover the dominant linear modes found are almost always those which seem likely to form bars 91 Formulation of the eigenvalue problem Kalnajs works in action angle variables in troduced in 423 As these are canonically conjugate coordinates he is able describe the dynamics using the elegant formulae of classical Hamiltonian mechanics eg Goldstein 1980 In particular the collisionless Boltzmann equation 1 can be written very compactly in Poisson bracket form 6F at FH70 30 where F is the distribution function and H is the Hamiltonian In order to calculate small departures from equilibrium we divide the Hamiltonian into a sum of the unperturbed axisymmetric part H0 plus a small perturbing potential h Similarly we divide the distribution function F F0 f Substituting into 31 expanding the Poisson bracket making use of the equations of unperturbed motion 3 i 6H0 i i i afIO i J i 7 awi 0 and w i ah i 9 31 and neglecting f h as being second order we obtain Elf 6f 7 3F0 6h 5 39m W m 32 to rst order We have adopted vector notation for the quantities to wnwa J Jr Ja and Q n a Since all unperturbed orbits are quasi periodic we can expand the perturbation potential h as a Fourier series in the angle variables 1 hJwt Wghm ljkmw 33 where m l The coef cients are given by 27r 27r hmJt hJ wte mquot d2w 34 0 0 We also transform f in the same way lntroducing these decompositions into 33 we see it must be satis ed by each Fourier component separately thereby converting the PDE 33 into a set of rst order ODEs 5fquot oF Wm9f11mm67 35 which has the solution t I fmJt lme mm hmJtequotquot399 dt 36 62 Sellwood and Wilkinson Assuming lfJ w tl does not grow any faster than 6 04 real and gt 0 we may write as a Laplace transform 1 ooioz I J 7 lwt 39 f wt 2W io m fJwwe do 37 With this substitution into 37 we can evaluate the t integral to obtain mJw 6F0 m Q 7 w 39 aJ39 This is an equation for one Fourier component of the perturbation to the DF caused by a single Fourier component of a disturbance potential having frequency w Notice that the denomina tor passes through zero for purely real to and certain values of Q J these are the familiar resonances where linear theory breaks down for steady waves as we saw in 42 However this problem does not arise for growing or decaying disturbance complex to The total response is the sum of these components fmltJ7W 38 1 1 lm J7 WWI 3E 7 39 M We obtain an eigenvalue equation for the normal mode frequencies u by requiring that the perturbation potential arises from the disturbance density Le vz u ww 4mm 4W5 3w wwd2 v 40 Two practical dif culties are posed by this eigenvalue problem Firstly the Fourier compo nents of the disturbance potential EmJw are coupled by Poisson7s equation Because of the rotational invariance of the Laplacian operator the different angular harmonics of the potential can be treated separately for in nitesimal perturbations but the entire set of l components for a given m remain coupled We therefore have to solve for eigenvalues of an in nite set of coupled equations Secondly it may seem that the use of action angle variables aggravates thedif culty of nd ing solutions to 41 since we have to evaluate the perturbed surface density 217 by integration over the velocities The transformation needed to express J wt in L1 coordinates can not be written in closed form except for a few special potentials Were we to eschew these variables and approximate the orbits as Lindblad epicycles we would avoid this problem but we would still be faced with the dif culty of calculating the disturbance potential through Pois son7s equation Since the solution of Poisson7s equation for an arbitrary mass distribution in a disc cannot be written down directly the eigenvalue problem is not much easier in z 1 coordinates Kalnajs 1977 proceeds by introducing a bi orthonormal set of basis functions the surface densities and corresponding potentials are related through Poisson7s equation and normalized such that 1 I I 7 hfgdz 1 lfzjv 41 27TG l 7 r 0 otherwise He then writes the unknown density perturbation 17 as a sum over the set 00 1 2 gagE where 17 7 hjEpdzr 42 Barred galam39es 63 The self consistency requirement for modes is now that both Sp and the perturbed potential h have the same expansion in the chosen basis which enables him to avoid calculating the perturbed density altogether With this trick he is able to rewrite the mode equation 41 in matrix form EMMWM an 43 j0 where the matrix coef cients are 1 6 h h MW y g m mg c121 lt44 they are more readily evaluated after integration by parts The overall technique involves a laborious calculation of all the coef cients ham which fortunately are independent of w and a non linear search for the eigenvalue w results have so far been obtained for three disc models Kalnajs 1978 Zang 1976 Hunter 199216 A number of short cuts have been devised to simplify the method such as using cold discs with softened gravity Erickson 1974 Toomre 1981 gaseous approximations Bardeen 1975 Aoki et al 1979 Pannatoni 1983 Lin and Bertin 1985 and moment methods Hunter 1970 1979 92 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 eigenfrequency in close agreement with the analytic prediction Zang and Hohl 1978 Sellwood 1983 Sellwood and Athanassoula 1986 A further example of such a comparison has been made in the simu lation shown in Figure 17 as usual the linear instability leads to a large amplitude bar The possibility that N body simulations somehow exaggerate the saturation amplitude and hence the signi cance of the instability was eliminated by lnagaki 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 see 10 for caveats 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 69 Sparke and Sellwood 1987 yet the majority of galaxies do not possess a strong bar Figure 3 To account for this we must understand how most galaxies could have either avoided this instability or subsequently regained axial symmetry We discuss several ways in which the instability might be averted in 95 but 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 16 The method has also been successfully used for spherical stellar systems Polyachenko and Shukhman 1981 Saha 1991 Weinberg 1991 64 Sellwood arid Wilkinson 93 Properties of the resulting bars The type of behaviour illustrated in Figure 17 is typical of almost every two dimensional sim ulation 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 eg 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 7 see 432 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 eg Figure 8 of Sellwood 1983 The gure rotation rate may quickly start to change as the model evolves further see 103 The rule that the initial bar nearly lls 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 198617 The most extreme axial ratios to survive for at least a few tumbling periods are in the range 4 7 51 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 pro les remarkably well 21 The stars within the bars in these simulations exhibit a systematic streaming pattern which again bears some resemblance to the observed velocity eld 251 94 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 ampli er 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 out wards ii trailing spiral waves that can reach the centre are re ected as leading waves iii a second re ection occurs at co rotation where a leading wave rebounds as an ampli ed trailing disturbance 17 A possible exception occurs for catastrophically unstable cold discs eg Hohl 1975 but as these discs are unstable on all scales down to the smallest that can be resolved one can argue that the local instabilities have caused so much heating that this is not a real exception to the ru e Barred galaries 65 Standing waves can occur at only those frequencies for which the phase of the wave closes which implies a discrete spectrum Super re ection off the co rotation resonance causes the standing wave to grow however making the mode unstable The super re ection which Toomre 1981 aptly named swing ampli cation was rst 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 95 Controlling the bar instability The mechanism for the bar instability just outlined indicates three distinct strategies for con trolling it These are i Raising the velocity dispersion to inhibit collective effects This option is unattractive as it would also inhibit spiral waves though it might be possible to construct a model galaxy which could sustain spirals in the outer disc and inhibit the bar instability by having an unresponsive region near the centre of the disc only lmmersing the galaxy in a massive halo as favoured by Ostriker and Peebles 1973 The additional central attraction increases the stabilizing contribution from rotational forces while keeping the destabilizing gravitational forces from the density perturbations unchanged There are two reasons why this method seems unattractive for galaxies rstly the halo must dominate the rotation curve even in the centre Kalnajs 1987 whereas evidence suggests that halos dominate only the very outer rotation curve van Albada and Sancisi 1986 and secondly galaxies stabilized in this way would exhibit multi armed spiral patterns Sellwood and Carlberg 1984 whereas the more common visual impression is of a dominant bi symmetric pattern lnterrupting the feed back cycle as favoured by Toomre 1981 The feed back loop will be broken if in going waves are prevented from reaching the centre This should occur if the wave encounters an ILR which damps the oscillation through wave particle interactions ILRs are likely to arise in galaxies having high circular velocities close to the rotation centre Lin 1975 was probably the rst to conjecture that a dense central bulge might be effective in preventing the bar instability A x i 39 96 Meta stability and tidal triggering The simulations reported by Sellwood 1989 were designed to test this last stabilizing mech anism He found that almost fully self gravitating discs could indeed be linearly stable when the centre was made suf ciently hard ie he found that no bar formed when initial perturba tions are kept to small amplitudes When the same model was subjected to a larger amplitude disturbance however it formed a bar with properties very similar to those in other linearly unstable models The reason the outcome depends upon the amplitude of the initial disturbance is that the Lindblad resonance can damp only weak perturbations particles become trapped in the valleys of a large amplitude wave 7 ie the resonance saturates Sellwood found that waves arising from noise uctuations in the random distribution of 2 gtlt 104 particles were of large enough amplitude to alter the linear behaviour Another method of inducing a bar in a meta stable galaxy model is by tidal triggering which was explored by Noguchi 1987 and by Gerin et al 1990 This work has been criticised 66 Sellwood arzd Wilkinson on the grounds that it is not fully self consistent Hernquist 1989 and Noguchi7s choice of Q 1 for the initial discs means that his simulations probably over estimate the ef ciency of bar formation Notwithstanding these criticisms this idea may offer an explanation for the possible over abundance of barred galaxies amongst binary pairs and small groups of galaxies Elmegreen et al 1990 97 Art alternative theory of bar formation While this section has focused on the idea 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 rst creating a bi symmetric density perturbation which would then exert suf ciently 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 Signif icantly 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 momen tum 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 Sell wood 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 Bell7s mechanism the instability is probably unrelated to the formation of strong bars 10 Evolution of the bar Many authors have described the bar as the end point77 of the simulation Sparke and Sellwood 1987 presented convincing evidence that a two dimensional stellar bar is indeed a steady robust feature when all particles except those in the bar are frozen and therefore prevented from interacting with it They knew at the time that the bar evolves through interactions with other responsive components but were unaware that an isolated bar can suffer a further instability when motion into the third dimension is permitted Barred galaxies 67 I 39 I 39 I 39 I 1 I I l 39 I 39 I 39 I I 39 l 39 I 39 I 39 I I 39 l 39 I I I 39 I I 39 I 39 I 39 I I I 39 I 39 I 39 I 39 I I 39 I 39 I 39 I 39 I I I 39 I 39 I 39 I I 39 I I I 39 I 39 I I 39 I 39 I I I I 39 I 39 I 39 I 39 I I 39 I 39 I 39 I 39 I I I I I I I I I I I I l j I I I I I A I I I I I I Figure 18 Pole and side views of bar formation and subsequent buckling in the fully three dimensional thicker disc experiment reported by Raha et al 1991 The side views are always shown from a position perpendicular to the bar major axis Only 2K of the 90K disc particles are shown in each plot and none of the 60K spheroid particles is shown The bar rotation period is a little less than 200 time units 10 Buckling instability The re hose buckling or corrugation instability for a stellar disc was considered by Toomre 1966 Kulsrud et al 1971 Fridman and Polyachenko 1984 and Araki 1985 in an idealized non rotating in nite sheet model Subsequent papers have all con rmed Toomre s original conclusion that buckling modes develop in this simple model only when the sheet is thin or more exactly when the velocity dispersion in the z direction is less than about one third Araki quotes 029 that of the velocity dispersions in the plane As the value observed in the solar neighbourhood is approximately 05 eg Woolley 1965 all these authors concluded that buckling modes are unlikely to be of importance in disc galaxies However the formation of a bar within the disc changes this conclusion Not only does a bar typically make the orbits more eccentric in the plane but the direction of their principal axes are also aligned ie the majority of stars in the bar move up and down the bar in highly eccentric orbits 49 The creation of organized streaming motion along the major axis of the 68 Sellwood and Wilkinson bar occurs without affecting the pressure77 normal to the plane7 and compromises the stability to buckling modes Raha et al 1991 Figure 18 shows the development of a bar in a disc which then buckles out of the plane the model illustrated is one of the two cases presented by Raha et al The viewing direction for the edge on frames is from a point always on the bar intermediate axis The bend develops soon after the bar has formed and becomes quite pronounced by time 16007 after which the bar regains symmetry about the plane but is somewhat thicker than before and has a distinctive peanut shape when viewed from the side The instability also weakens the bar amplitude in the plane only slightly in this case7 and could perhaps destroy it entirely when very violent Raha et al 1991 A similar outcome was originally reported by Combes and Sanders 1981 and later con rmed and strengthened in simulations of much higher quality Combes et al 1990 However7 these authors propose that the bar thickens because of the narrow instability strip associated with the vertical Lindblad resonances within the bar Pfenniger and Friedli 1991 attempt to develop this alternative mechanism7 which has to be reconciled with their empirical results that collective effects and a z asymmetry in the potential are essential While there is no doubt that a narrow strip of vertically unstable orbits exists in their N body bar models7 it seems unlikely that the comparatively small fraction of orbits affected at any one time could lead to the exponentially growing7 uniform bend7 of the type shown in Figure 18 On the other hand7 Pfenniger and Friedli 1991 and Raha 1992 show quite convincingly that the nal peanut shape of the bar is caused by a large number of orbits trapped about the stable 22a1 periodic orbit family 523 thus the structure of bar after the instability has saturated7 though not the linear instability itself7 is determined by the periodic orbits 53 Moreover7 the existence of the re hose instability in thin non rotating bars was predicted by Fridman and Polyachenko 1984 The N body simulations of Merritt and Hernquist 1991 con rmed that highly prolate models suffer from buckling modes7 which clearly cannot be caused by unstable orbits in their Stackel initial models It therefore seems most likely that the buckling modes in both these and the rapidly rotating bars are of the re hose type 102 Peanut growth The nal bar in Figure 18 resembles a peanut shape when viewed from the side A similar shape appeared in every bar which formed in the many experiments reported by Combes at al 19907 although it is not certain that the formation mechanism was the same Their results seem to indicate that any strong bar which forms in a galaxy must quickly become much thicker than the disc7 a conclusion at variance with those observers who argue that bars seem to be thin 69 Kormendy 1982 The simulators have7 of course7 seized upon box peanut shaped bulges sometimes observed 233 and argued that they are the thickened bars Two further pieces of indirect evidence add credence to this speculation rstly7 it is believed on admittedly very slender evidence that the rotation rate of the stars in a box peanut shaped bulge is independent of distance from the plane7 a property which is not found for normal bulges Kormendy and lllingworth 1982 Secondly7 the fraction of box peanut shaped bulges in nearly edge on galaxies is some 20 69 Shaw 1987 which is not very different from the expected fraction of strongly barred galaxies7 when allowance is made for those systems in which the randomly oriented major axis of the bar must lie close to our line of sight Barred galasies 69 Unfortunately there are two drawbacks to this simple association Athanassoula private communication points out that most though not all box peanut bulges are much less exten sive relative to the host galaxy7s size than are strong bars Moreover Shaw private commu nication raises the further concern that the 30 of strong bars expected in edge on galaxies have apparently not all developed a recognisable peanut shape A further observational test of this hypothesis might be to determine whether the luminosity functions of bars and boxpeanut bulges are similar though dust obscuration might make this dif cult Before this last question acquires the status of a major puzzle however it seems necessary for the simulators to verify that the bars continue to thicken in a much wider range of mass distributions than have so far been tested especially those having a more realistic bulge with a high central density 103 Continuing interactions with the disc Sellwood 1981 emphasized that when the initial bar in his simulations was short compared with the total extent of the disc it continued to exchange angular momentum with the outer disc through further spiral activity As always the trailing spirals remove angular momentum from particles at their inner end Sellwood found that this enabled more stars to be trapped into the bar which increased its length and also lowered its pattern speed In the most extensive disc he studied this process continued through a number of spiral episodes while the bar increased in length by more than 50 Interestingly the changes in both bar length and pattern speed conspired to keep co rotation just beyond the end of the bar Combes and Sanders 1981 on the other hand reported that bars tend to weaken in the long term There are at least two physical reasons for their different nding rstly their models started with much less extensive discs precluding the type of secondary bar growth Sellwood observed secondly theirs were three dimensional models in which the bar thickened normal to the plane which caused the strength in the plane to decline see 101 No three dimensional models with an extensive outer disc have so far been run to settle the question of which effect w1ns 104 Interations with spheroidal components The large majority of simulations to date have employed xed additional contributions to the radial force to represent the e fects of spheroidally distributed matter such as the bulge and halo This computational expedient excludes the possibility of interactions between the disc population and a responsive bulgehalo component In a direct check on the validity of this approximation Sellwood 1980 found it adequate while the disc remained nearly axisymmetric Once a strong bar had formed however the initially non rotating bulgehalo population of particles began to gain angular momentum from the disc at a prodigious rate Thus the coupling between the disc and spheroidal populations of particles in barred galaxies ought not to be ignored The interaction process can be described as a kind of dynamical friction Tremaine and Weinberg 1984a between the rotating massive object the bar and a background sea of col lisionless particles Weinberg 1985 calculated that angular momentum should be exchanged between the bar and a non rotating spheroid at a rate consistent with Sellwood7s observed value Unfortunately Sellwood7s experiment did not continue for very many bar tumbling periods and did not reveal the long term consequences for both the bar and the halo population of a very protracted period of interaction We therefore still do not know whether the bar pattern speed 70 Sellwood and Wilkinson will be reduced or even raised7 whether its amplitude will be increased or diminished or how the spheroid will appear once a substantial exchange of angular momentum has occurred The only truly long term experiment to examine this issue has been conducted by Little and Carlberg 19917 but for computational economy they again used a two dimensional model Their at halo77 was an entirely pressure supported population of particles co planar with the rotationally supported disc in which the bar formed They found that the bar was indeed slowed by the angular momentum loss to the halo but a more realistic experiment is clearly warranted The predicted ef ciency of angular momentum transfer is embarassingly high7 suf cient to arrest a rigid bar in a few rotations7 yet a number of lines of evidence suggest that bars in galaxies rotate rapidly We are not7 however7 forced to an unattractive conclusion that bars must either be very young or they have stopped rotating there are other ways to avoid this dilemma The most plausible seems to us that Weinberg over estimates the ef ciency of exchange by assuming a non rotating spheroid The luminous bulges of galaxies generally rotate suf ciently rapidly for their attening to be consistent with a rotating isotropic model 69 Kormendy and lllingworth 1982 In the few cases that have been measured7 bulge streaming velocities are typically 40 of those at the corresponding distance in the disc and the angular momentum vector lies in the same direction as that of the disc This degree of rotation will alter the angular momentum exchanges with a bar7 and may make them less ef cient It is not clear whether the observed angular momentum has been acquired from the disc component already7 or whether it was primordial Weinberg himself also notes that the bar is simultaneously spun up by interactions with the outer disc7 and speculates that it could act as a catalyst transferring angular momentum from the disc to the spheroid while not changing its own angular momentum content by much Finally7 the bar is not a rigid body and probably has a much lower moment of inertia possibly even negative than that assumed by Weinberg These issues can be addressed only by high quality7 long term experiments 105 Destruction of bars Even three dimensional bars seem to be long lived Pfenniger and Friedli 1991 report that their bar models cease to evolve after the thickening phase7 and survive essentially unchanged as long as the calculation is run Thus although the buckling instability 101 dented the common perception that bars are rugged structures7 it does not seem to have invalidated it It is therefore interesting to ask whether a galaxy7 having once acquired a bar7 could ever rid itself of it A few ideas for achieving this have been mooted The most recent Raha et al 1991 is that the buckling instability could be so violent that the bar is destroyed Raha et al seemed to nd that the buckling instability was more violent when the bar formed in a thinner disc7 though in both their cases the bar was merely weakened7 not completely destroyed If ever the buckling instability could be violent enough to destroy the bar7 therefore7 the bar would have to have formed in an extremely thin7 and probably very young7 disc Another suggestion is that bars could be destroyed during an interaction with a companion galaxy It has generally been felt that an interaction violent enough to destroy the bar would also destroy the disk7 but Pfenniger7s 1991 report of a simulation in which a dwarf was accreted by a barred galaxy seems to show that the bar can be destroyed7 and the companion disrupted7 without doing very much damage to the disc Barred galam39es 71 A further suggestion has been made by Norman and his co workers Pfenniger and Norman 19907 Hasan and Norman 1990 that the growth of a comparatively light7 but dense7 object at the centre can destroy the bar They argue that the growth of central mass changes the major orbit families which support the bar7 and thereby threaten its survival Friedli et al 1991 present preliminary results which seem to indicate that a bar can drive suf cient gas towards the galactic centre to have this effect 7 Le that bars in gas rich galaxies may self destruct 11 Conclusions It should be clear that we are still far from a complete understanding of the basic structure of barred galaxies7 but progress has been rapid in recent years 7 especially where orbit studies have been connected to observational or simulated results There are a few aspects where the observed facts seem to t reasonably well with theoretical ideas On the bright side7 it seems very likely that strong bars are formed by the dynamical instability discussed in 9 Bars formed in the N body experiments end near co rotation and have mass distributions and kinematic properties which seem to correspond with those observed7 though more detailed comparisons would be desirable Moreover7 such bar models appear to have the right strength7 and to rotate suf ciently rapidly to shock gas in places resembling the dust lane patterns in some barred galaxies The ow patterns described in 6 are understood in terms of orbit theory and t the observed kinematics for entirely plausible model parameters We also think we understand the origins of rings 7 suf ciently well to be able to interpret them as signatures of major resonances with the bar pattern These successes add up to a compelling7 but indirect7 case for a bar pattern speed in galaxies which places the major axis Lagrange point just beyond the bars end Such a value is consistent with the more direct observational estimates of pattern speed 26 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 ln particular7 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 Moreover7 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 con rm 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 trapped7 or semi trapped7 about the main family of orbits aligned with the bar the 1 family Apart from this7 knowledge of the main orbit families has improved our understanding of gas ows 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 progress7 of a kind7 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 this7 our fragmentary understanding of the evolution of barred galaxies7 and almost total blanks on the origins of lenses and the pronounced asymmetries in some galaxies7 seem of secondary concern 72 Sellwood and Wilkinson Probably the most pressing need on the theoretical side is for a more sustained attack on the orbital structure of three dimensional bars7 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 deter mines the 30 fraction of galaxies seen to have strong bars Since the bar instability seems able to create strong bars in nearly all disc galaxies7 how can we account for the current mod erate fraction The ideas for controlling the bar instability 95 and those for destroying bars 105 do not explain either why some 70 of galaxies manage to avoid a bar instability7 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 bars7 or is some totally different mechanism required The bar instability we discuss in 9 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 review7 but it does seem likely that the bar instability would be profoundly affected by the manner in which galactic discs form Sellwood and Carlberg 19847 in a few preliminary experiments mimicking gradual disc formation7 found that the velocity dispersion of the stars rose suf ciently rapidly to inhibit the formation of a strong bar as the disc mass built up Further experiments of this kind7 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 distinct7 but we have seen in 10 that there may be no dynamical basis for considering them as separable Moreover7 as it seems likely that the bar formed from the disc7 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 objects7 we should be encouraged that progress over the last few years has been rapid7 especially since the knowledge which enabled us to formulate many of these questions has only recently been acquired The many new telescopes and observational techniques7 particularly operating in the infra red7 are likely to advance the subject at its currently intense rate Acknowledgments We would like to thank J Binney7 D Earn and D Merritt7 as well as E Athanassoula7 A Bosma7 R Buta7 G Contopoulos7 D Friedli7 J Gallagher7 D Pfenniger7 M Shaw7 L Sparke and P Teuben for comments on the manuscript We acknowledge support for brief visits to ST Scl and Manchester as part of the effort to propel this review forward References 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