Astron. Astrophys. 344, 483-493 (1999)

3. Mass redistribution within a barred galaxy

We will use the simulation presented in Tables 1 and 2 as an illustration for our detailed discussion on mass redistribution within a barred galaxy. This simulation starts with a smooth disc developing spiral structure after some 8000 timesteps. The bar forms at and evolves quickly for a few revolutions. The subsequent evolution is more modest; the bar grows slowly in length while its pattern speed decreases.

3.1. Changes in the surface density profile

The mass distribution within the disc stays very close to the initial one until the bar starts to form. We can see this in Fig. 1 where the azimuth-averaged surface density profile, , is plotted for four timesteps in our simulation. The curves representing , the initial state, and , the disc prior to bar formation, are very similar and almost coincide. The only deviation of the curve is found in the radial range  l.u. and is due to the emerging spiral pattern. The arms have grown stronger and extend further out ( l.u.) in the galaxy by timestep 10 000. The bar then forms just prior to the last surface density profile shown in Fig. 1 (). Except for the disturbances due to the spiral arms, all the curves in Fig. 1 closely follow the theoretical profile of the Kuzmin disc, although the low number of stars at large radii makes them somewhat noisy.

Fig. 1. The surface density profile, , of our simulation at three timesteps prior to bar formation. The initial Kuzmin disc is shown as a thick grey line. Timestep 10 400 shows when the bar just has formed.

After the bar has formed at the surface density profile takes on a different shape (Fig. 2). The profile becomes exponential in the bar region which extends out to between 20 and 25 l.u. (The bar will slowly increase its length to follow the co-rotation radius when it gradually slows down as indicated in Table 2 ; see also Lerner et al. 1999.) The slope of is steeper than for the original Kuzmin disc in this area. Outside the bar region comes a plateau extending out to  l.u. This plateau is also roughly exponential and decreases very slowly with radius before it starts falling off faster again further out. This outermost region seems not to be exponential but retains a Kuzmin-like profile. (We will return to the shape of this region in Sect. 3.4.) In the curve representing , there is a small peak at  l.u. indicating the presence of spiral arms carrying material outwards. As time passes, the surface density goes down in the plateau region outside the bar, while it continues to increase further out in the galaxy. The net result is that decreases with time in a large part of the inner galaxy ( l.u.), while it increases in the outer part of the disc ( l.u.) and in the core of the galaxy ( l.u.). The general trend once the bar has formed is thus that material in the outer parts of the galaxy starts migrating outwards, while matter in the inner parts migrate inwards.

Fig. 2. The surface density profile, , of the simulation at four timesteps after bar formation. The initial Kuzmin disc is shown as a thick grey line.

In the simulation we present here, the border between the regions with inward and outward migration occurs at a radius of approximately 10 l.u. (Fig. 3). This radius is less than half the length of the bar and coincides with the turn-over of the rotation curve (which decreases from 11.5 l.u. to 10.6 l.u. through the simulation). Throughout the simulation, 40% of the disc mass stays inside this radius and the other 60% stays outside it. This is only true in a statistical sense, of course, since most stars in the bar have highly elliptical orbits that will cross this border several times during each revolution.

Fig. 3. The evolution of different radii containing a certain fraction of the disc mass.

Is it possible that the co-rotation radius also acts as a dividing line in our simulation? This turns out not to be the case; 62% of the disc mass is located inside the co-rotation radius at and this value falls to 59% by the end of the simulation, although the co-rotation radius by then has moved more than 5 l.u. further out in the disc. Most of this mass loss occurs during the first 1500 timesteps when the bar still does not extend all the way out to its co-rotation radius. Hence, we believe that the effect is caused by stars not yet bound to the bar being perturbed outwards in the disc.

A clue to the border at 10 l.u. is found when following individual stars through the simulation. Almost no stars (%) starting out with  l.u. will ever leave the bar region, while stars from larger radii can end up anywhere in the galaxy, although most of them avoid the central part of the bar. It appears that the potential gradient at  l.u. is small enough to allow the early spiral pattern and the proto-bar to perturb a fraction of the stars outside that radius outwards.

3.2. Mass loss and mass gain in annular regions

The gain or loss of mass in different annular regions in the galaxy is shown in Fig. 4. That the mass distribution does not deviate from the initial one for a long period before the bar forms is clearly evident. The first minute changes can be seen at when the first signs of a spiral structure appears in the simulation. The spiral arms grow in strength and trigger the bar formation process. The bar forms at and by its shape is already quite similar to its appearance at , the end of our simulation.

Fig. 4. The mass change in seven annular regions of our simulation. No change can be seen before the first appearance of a spiral structure approximately 2500 timesteps before the bar forms.

From Fig. 4 we learn that the bar formation is associated with a dramatic reshuffling of matter within the disc. In the inner parts, a new semi-stable mass configuration is reached within a few bar rotation periods, while the outer parts of the galaxy need Gyrs to adapt to the new conditions.

The very centre of the disc ( l.u.) experiences a clear gain of mass, as we already have seen in Fig. 1. At the end of the simulation, this gain amounts to an increase of about 30% of the original mass within this region. In the rings with  l.u. and  l.u., we instead find a loss of stars of about 20% and 35%, respectively. The major loss of mass, however, occurs in the region  l.u., i.e. the region just outside the bar region. Although there is a brief period without any mass changes lasting for some 600 timesteps just when the bar forms, it is followed by a rapid loss. Contrary to the regions inside  l.u. where most of the mass changes take place within a few bar rotation times, the mass loss in the  l.u. region continue at a high rate for a long time. At the end of the simulation, the number of stars populating this region is down to one-third of the original value.

Continuing further out in the disc, we find that the region  l.u. experiences a fast growth when the bar forms. It reaches a peak 700 timesteps later when the population in the region has been increased by two-thirds. However, this over-population will go away almost as quickly as it was built up. Some 1800 timesteps later the population is back to its initial level and the decline then continues until the end of the simulation, when the number of stars is down by 35%.

A similar trend with an even sharper rise followed by a decline, albeit a more moderate one, is seen in the region  l.u. The peak is reached at a later timestep, clearly indicating that we are witnessing a wave of stars moving outwards in the galaxy. The decline is more gentle than for the  l.u. region. This is partly, but not entirely, an effect of the larger size of the  l.u. region. If we would have subdivided this region into two, we would have found that the inner part ( l.u.) has a steeper decline than the outer part, but none of them is as steep as the decline for the  l.u. region.

The picture is further strengthened when considering the region  l.u. which continues to gain stars for several thousand timesteps, finally reaching a density some 150% above the initial value.

3.3. Stellar redistribution

To understand the trends regarding mass gain and mass loss at certain radii better, we have also studied the fate of stars in different annular regions. As we have already discussed, almost all stars starting out with  l.u. end up in the bar. Hardly surprisingly, the smaller the initial orbit is, the more central in the bar will the final orbit be (Fig. 5). Most stars in the region  l.u. also contribute to the bar, forming its outermost layer, but stars from this region can also be found spread out in the galaxy as can be seen in the third frame of Fig. 5. Of the stars with initial radius  l.u. not very many contribute to the bar and those which do are found in the outermost orbits of the bar. In the fourth frame of Fig. 5, we see them pile up at the end points of the bar where they spend most of their time.

Fig. 5. The final location of the stars in the annular regions studied in Fig. 4. The original regions are indicated by the circles. Note that the scale is increased by a factor of 4 in the lower row of frames and that the region  l.u. appears twice with different scales. These diagrams are based on a randomly selected sub-sample consisting of 100 000 stars.

Stars starting out with  l.u. are instead widely distributed through the final disc as can be seen in the fifth frame of Fig. 5. Here, we have decreased the scale by a factor of 4 to demonstrate the fairly uniform distribution these stars end up in. Superimposed on this distribution are the two previously mentioned clumps at the end points of the bar. Stars initially having  l.u. are less affected by the bar and stars from an initial annular region also end up in a ring. This ring is, however, much wider than the original region due to disturbances from the bar as well as the spiral arms. From stars originating in the region  l.u., this ring is fairly broad (frame 6 of Fig. 5) and we can also see a smaller number of them clumping up at the ends of the bar. No such clumping is evident for the region  l.u. Most of the stars in this region stays within the region and if they leave it, they move outwards in the galaxy. Continuing further out to the region  l.u., the tendency to leave the original region has decreased considerably. The dispersion clearly goes down with increasing radius and at the outer rim of the disc ( l.u.), a star will at the end of the simulation be found less than 15 l.u. away from its original orbit.

3.4. Stars ejected from the disc

From the discussion in the previous subsection, we would not expect any stars outside  l.u., since the outermost stars located at  l.u. are not disturbed into orbits with larger radii than this value. Yet, when we study the 100 000 stars for which we have data stored, we find 3805 of them, or nearly 4%, at  l.u. at the end of the simulation! A few of them (91 stars) have even tried to leave the coordinate system, attaining distances of 500 l.u. (see Fig. 6)! Where do they come from and what process have managed to throw them out to such large distances?

Fig. 6. The final positions of 100 000 randomly selected stars in our simulation. All stars with  l.u. (3805 stars) are plotted with large symbols. The 91 stars that tried to leave the polar coordinate grid (indicated by the circle) are marked with larger symbols. The stars with  l.u. have been reduced to 25 000 to improve the appearance of the central region.

To answer the first part of that question, we have selected all stars with  l.u. in timestep 20 000 (Fig. 6) and plotted their initial positions (Fig. 7). All the expelled stars seen in Fig. 6 are included in Fig. 7, except eight which actually come from the outer rim of the original disc. As we can see, they do not start out from random locations but all come from the region  l.u.

Fig. 7. The initial positions of the same stars shown in Fig. 6. All the 3805 far-out stars marked in Fig. 6 can be seen in this figure except 8 which come from the outer rim of the galaxy. Note that the scale in this figure is 10 times larger than in Fig. 6.

In Figs. 2, 4 and 5, we showed that this region loses a lot of its stars and that they migrate outwards in the disc. Obviously, this migration is far more than just a gentle shift of the orbits a slight distance. Most of the expelled stars in Fig. 6 have increased their distance from the centre by 10 to 20 times. These stars are not on circular orbits, of course, since there is no mechanism capable of circularizing orbits far out in the galaxy. Instead, they travel on highly elliptical orbits reaching a distant apocentre before turning around and diving back towards the pericentre in the inner parts of the disc. The total number of stars with apocentres larger than 150 l.u. is thus higher than 3.8%; when we add together all stars that during some period have been outside  l.u. the number goes up to 6.7%. If we look particularly at stars from the region  l.u., we find that more than 20% of them have been at  l.u. sometime during the simulation.

We thus expects the outer part of the disc to consist of two dynamical populations of stars: those which originated there and travel on more circular orbits, and those which come from the inner parts of the disc and move on highly elliptical orbits. That this really is the case can be seen in Figs. 8 and 9 which show the tangential and radial velocities, respectively, of 25 000 stars at a few selected timesteps.

Fig. 8. The tangential velocities of 25 000 randomly selected stars from our simulation at four different timesteps. The first frame shows the velocities when the bar just has formed.

Fig. 9. The radial velocity component of the stellar velocities shown in Fig. 8.

The first frame of Fig. 8 shows a timestep when the bar just has formed. For  l.u. the velocity dispersion has broadened the original smooth velocity curve markedly, while stars at  l.u. still are undisturbed by the new-born bar and the spiral pattern that preceded it. The dispersive action of the spiral pattern making its way outwards can easily be seen in the subsequent frames.

The depopulation of the region  l.u. discussed in Sect. 3.2 is clearly visible if we compare timesteps 10 400 and 20 000 of Fig. 8. The massive wave of stars heading for the outer parts of the galaxy can be seen in timesteps 11 500 and 12 600 as a band of stars stretching out below the original velocity curve. In the third frame, we can even see that this wave precedes the spiral wave at radii larger than 100 l.u.

The process is evidently fast. The first stars cross the  l.u. border about 2400 timesteps after bar first appears, corresponding to a period of only 1.2 Gyr. Less than 0.5 Gyr later, we find stars that started out at circular orbits around  l.u. at 10 times this distance! The trends of fast mass relocation seen in Fig. 4 certainly holds true even for the outskirts of the galaxy!

With the knowledge of the two dynamical populations, we will now return briefly to the question whether the outermost of the three regions found in Fig. 2 is exponential or Kuzmin-like. The problem is that the curve () in Fig. 2 is rather flat in the region  l.u. and can be fitted quite well both with an exponential function and a Kuzmin profile, although the Kuzmin fit is the better of the two. To achieve a better result, it would be preferable to have a larger region to study. Since stars in the original population start to drop off beyond  l.u. we can not make such a fit to that population. The second population, however, extends much further out and with the aid of Fig. 8, we can separate this population from the other. We have thus studied its surface density profile in the region  l.u. and found that the ejected population clearly is Kuzmin-like.

From Fig. 9, it is evident that there is an asymmetry in the radial velocity distribution caused by the outward streaming. There are more stars with large positive radial velocities than negative ones at timesteps 11 500 and 12 600. The asymmetry still persists at , although it is much smaller since there are many more stars falling back from their highly elongated orbits by then.

The fact that we do not see any concentration of stars with at radii  l.u. clearly shows that this region no longer contains stars on circular orbits. The stars that do show up in the diagram with are actually stars passing through the pericentres of their elliptic orbits.

This result has important implications for observers of barred galaxies. When a bar is present, we can not assume that stars observed at a certain radius have been born at that radius. Instead, we should expect to see a mix of stars born at this radius and stars born just outside the bar region. Due to this mixing of stars throughout the disc, gradients in stellar properties and chemical abundances ought to be smaller in barred galaxies than in non-barred systems. That this really is the case, at least regarding chemical abundances, is known from several studies. It has, for example, been shown that the O/H-gradient across the disc is less steep in barred galaxies than in comparable non-barred systems (Vila-Costas & Edmunds 1992; Zaritsky et al. 1994; Martin & Roy 1994).

What really matters for observers who want to detect these two components directly is, of course, what the projected velocities look like. In Fig. 10, we have therefore plotted the projected velocities of the last timestep (timestep 20 000) seen from three different angles (parallel with, inclined 45^o to and perpendicular to the bar) together with the projected velocities from the original axisymmetric disc (timestep 8000). Although the two components no longer are as clearly separated as in Fig. 8, they can still be seen easily, especially at distances larger than 80 l.u. from the centre of the galaxy. Stars belonging to the ejected population can also be seen as a sparse band around even entering into the two "forbidden" quadrants (upper-left and lower-right).

Fig. 10. The projected velocities of 25 000 randomly selected stars from our simulation. The first frame shows the projected velocities of the original axisymmetric disc, while the other three frames show the final timestep from three different viewing angles. The observer is assumed to be in the plane of the disc.

It is encouraging to note that the velocity diagrams look very similar in their outer parts regardless of the orientation of the galaxy. This is, of course, what we could have expected from the axisymmetric distribution of ejected stars seen in Figs. 4 and 6. The existence of two dynamical populations clearly produces a large velocity dispersion throughout the galactic disc. We thus suggest that high-resolution spectroscopy could be used to reveal the existence of these two dynamical stellar populations in the outer parts of barred galaxies.

The velocity difference between the two populations as seen in the outer parts of the diagrams in Fig. 10 is about 0.1 v.u. (velocity units ). If we assume that the maximum rotational velocity in the disc is 250 km s^-1, 0.1 v.u. would correspond to roughly 60 km s^-1. Allowing for an inclination of up to 60^o from the line-of-sight would still yield a velocity difference of 30 km s^-1, which should be no problem to measure if a bright enough tracer of the two populations can be found.

Far out in the disc, the two populations should be discernible as two distinct peaks in the spectrum. Closer to the centre, the two peaks will blend together. However, the wing of "forbidden" velocities will always be present and we suggest that observers trying to find the second population should search for this wing.

Two synthetic spectra calculated from our simulation are shown in Figs. 11 and 12. Both figures contain four spectra: one of the original axisymmetric disc and three of the disc at (parallel with, inclined 45^o to and perpendicular to the bar). The slit used is 10 l.u. wide and the spectral resolution adopted is 0.01 v.u. In Fig. 11, the slit was positioned at  l.u. The presence of the two populations can clearly be deduced from the jump in the profiles at  v.u. and the broad wing extending down to -0.15 l.u. In Fig. 12, the slit was located further out ( l.u.) and not only the broad wing is seen but also two peaks corresponding to the two populations.

Fig. 11. Four synthetic spectra taken with a 10 l.u. wide slit at  l.u. having resolutions of 0.01 v.u.

Fig. 12. Four synthetic spectra taken with a 10 l.u. wide slit at  l.u. having resolutions of 0.01 v.u.

The main problem is what type of objects to observe. Ideally, absorption lines of older stars should be observed. Early-type stars (O-, B- and A-stars) are not suitable for such studies since they have too short life-spans. If such stars are found in the outer part of the disc, they must have been born there, and their motion will thus reflect the dynamics of the local interstellar medium and not the ejected stellar population. Emission lines from HII -regions would also be a bad choice for the same reason. Emission lines from planetary nebulae, however, would be suitable since they will share the dynamics of the stellar populations.

3.5. Ejection mechanisms

Let us now turn our attention to the question of the mechanism responsible for ejecting stars to the outer parts of the disc. In the leftmost frame of Fig. 13, we show one of these stars being flung out into increasingly larger ellipses. In the middle frame of the same figure, we have plotted how the distance from the galactic centre evolves with time as the ellipses gets more and more eccentric. The angular momentum of the star is superimposed on this frame. As is clearly evident from this figure, the pericentre distance stays roughly the same, around 20-25 l.u., while the apocentre increases dramatically for each orbit. To understand this strange behaviour, we have to study the orbit in a reference frame co-rotating with the bar, i.e. a frame in which the bar appears to have a fixed orientation. Such a diagram is shown in the rightmost frame of Fig. 13 superimposed on an image of the bar at the final timestep of the interval shown.

Fig. 13. The trajectory during 5400 timesteps of a star being ejected to the outer part of the galaxy. The leftmost frame shows the trajectory superimposed on an image of the galaxy at the last timestep shown. The end-point is marked by a large dot. The middle frame shows how the star's distance from the centre of the galaxy (thick line) varies with time during the selected time period. A plot of the star's angular momentum (in arbitrary units) is also shown in the middle frame (thin line). The rightmost frame shows the trajectory plotted in a coordinate system co-rotating with the bar. Again, the trajectory is superimposed on an image of the bar at the last timestep shown. Only 4000 of the stars used in the simulation are included in the background images.

Although the galaxy rotates counter-clockwise in the simulation, the sense of motion in the rightmost frame of Fig. 13 is clockwise, since stars outside the bar region will have a lower rotation speed than the bar. At the selected star starts out at coordinates , travels 270^o clockwise in a wide arc around the bar and then performs a small loop at the lower end of the bar before temporarily leaving the diagram after having completed a full revolution around the bar. It then reappears and make three additional loops in the vicinity of the bar during the following 5000 timesteps.

These loops are the pericentre passages as seen in this co-rotating reference frame. As we can see, all four of them appear either in the top-right or the bottom-left quadrant, and this is the crucial clue to the problem. If the pericentre passage occurs in either of these two quadrants, the star will travel behind the bar during the time of its closest approach to the centre (since the galaxy is rotating in the counter-clockwise direction). The star will thus experience a torque that transfers angular momentum from the bar to the star. The gain in angular momentum will throw the star into a more eccentric orbit taking it farther out in the galaxy. If, as for the star in Fig. 13, several consecutive pericentre passages all happen to occur in these "accelerating" quadrants (hereafter called -quadrants), the apocentre evidently can be moved far outside the galaxy in just a few orbits!

However, if this model is correct we would expect to see the opposite effect when a star makes a pericentre passage in one of the other two quadrants (decelerating or -quadrants). The star would then be leading the bar and would thus experience a braking force that reduces its orbit. That this indeed is the case is shown in Fig. 14.

Fig. 14. In this figure, similar to Fig. 13, the trajectory of a star both gaining and losing angular momentum in close encounters with the bar is shown. The evolution of the star's orbit is followed for 9000 timesteps. The pericentre passages have been numbered for the discussion in the text.

The middle frame of Fig. 14 shows the distance from the galactic centre for the selected star, as in the previous figure. As we can see, the apocentre distance of this star varies both inwards and outwards with time. We have numbered the pericentre passages in the distance diagram and the corresponding loops in the co-rotating frame to provide a better understanding of the events.

The first pericentre passage occurs in an -quadrant and thus increases the apocentre of the orbit. The following three pericentre passages, however, all occur in the -quadrants and the apocentre distance consequently decreases for each of these three orbits. After the last deceleration loop, the star actually has lost so much angular momentum that its orbit is smaller than the one at the beginning of the time interval shown. The star will, however, not stay in this orbit, since the subsequent three pericentre passages are accelerating ones throwing the star outwards again.

From Fig. 14, we can also learn that it not only matters in which quadrant the pericentre passage appears but that the angle to the bar also is of importance. Loops located almost perpendicular to the bar (such as loops no. 2, 3 and 6 in Fig. 14) will have a small effect on the orbit, while loops closer to the ends of the bar (loops 1, 4 and 5) will affect the orbit more dramatically, as clearly is evident from Fig. 14. We also expect the pericentre distance to be of importance; the closer to the bar, the stronger will the influence from the bar be.

To investigate the importance of the angle to the bar, , we have calculated how the torque () a star at  l.u. experiences from all the stars within  l.u. varies with (Fig. 15). As expected, the torque is negative in -quadrants and positive in -quadrants. The force from the bar is symmetric when the star is parallel or perpendicular to the bar and is then zero. The changes in angular momentum, , to the star is given by

The major contribution to this integral will come from a short period of time around the pericentre passage, since declines rapidly with increasing r. This explains why we see rather abrupt jumps in the angular momentum curves in Figs. 13 and 14 at the times of closest approach. The size of will thus mainly depend on the -angle of the pericentre.

Fig. 15. The torque a star at  l.u. experiences from all the stars with  l.u. at the end of the simulation as a function of its angle, , from the bar major axis.

From Fig. 15, we learn that the maximum can be expected for stars making their closest approach at . The value of is more sensitive to the closer to the end points of the bar the star gets. Studying the angular momentum of the star in Fig. 14 during its seventh pericentre passage, we find that, at first, it rises steeply but then it declines equally rapidly, although not all the way down to its pre-passage value. This is not surprising since the pericentre loop is located at the end of the bar, starting in an -quadrant but then shifting over to a -quadrant. The loop is, however, slightly offset towards the -quadrant, thus producing a net gain in angular momentum.

We have now got a natural explanation for the depopulation of the region just outside the bar area ( l.u.) seen in Figs. 2, 4 and 8. Stars in this region will inevitably have their orbits changed each revolution due to the strong perturbing forces from the bar. Any circular or slightly elliptic orbit will quickly be perturbed into a high-eccentricity orbit that will change continuously. We can now understand why stars originating in the region  l.u. end up so evenly dispersed through the galaxy as we have seen in frame 5 of Fig. 5. If such a star is not captured by the bar it will be flung into a randomly oriented high-eccentricity orbit. Stars experiencing several consecutive acceleration passages may even receive enough energy to escape from the galaxy. Even stars which do not attain the proper escape velocity will travel so far away from the galaxy that they easily can be perturbed onto escape trajectories by galaxies in the neighbourhood during encounters that otherwise would not be violent enough to expel stars into the intergalactic space.

The area of depopulation extends from about the co-rotation radius past the outer Lindblad resonance (located at 33 l.u. at the end of our simulation). This is consistent with results from analytical potentials, which also predict that these stars will end up in highly elongated orbits with some of them even being ejected from the galaxy (Contopoulos 1981).

Recent studies (Theuns & Warren 1997; Ferguson et al. 1998) have found direct evidence for an intergalactic population of stars in the Fornax and Virgo galaxy clusters. It is still very uncertain how large the fraction of unbound stars is, but conservative estimates suggest that at least 10% of all the stars in the clusters are intergalactic and the value may be as high as 40%. Current theories predict that these stars either are left-overs from the time when the clusters formed or have been ejected from the galaxies during close encounters and collisions between galaxies. With the results from this paper, we propose an additional mechanism capable of ejecting stars from a normal barred galaxy not involved in violent encounters.

Could this mechanism be an important source for the intergalactic population? Probably not, if the higher mass estimates turn out to be right. If we make the assumption that all stars reaching  l.u. are perturbed into intergalactic space, our simulated galaxy has still only lost 3% of its total mass over a 5 Gyr period. If we further take into consideration the fact that only a fraction of the galaxies in a cluster are barred galaxies and the fact that real galaxies have a flatter rotation curve than our simulation, it seems even less likely that barred galaxies are a major contributor. The largest fraction of bar-ejected stars are probably to be found in the outer parts of galaxy clusters, where barred spirals are more abundant and collisions occur less frequently than in the centre of the clusters.

© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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