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arXiv:astro-ph/0302198v1 11 Feb 2003

Mon. Not. R. Astron. Soc. 000, 1–11 (2001) Printed 2. Februar 2008(MN LATEX style file v1.4)

Orbital dynamics of three-dimensional bars:

IV. Boxy isophotes in face-on views

P.A. Patsis,1Ch. Skokos,1,2E. Athanassoula3

1Research Center of Astronomy, Academy of Athens, Anagnostopoulou 14, GR-10673 Athens, Greece

2Division of Applied Analysis, Department of Mathematics and Center for Research and Application of Nonlinear Systems (CRANS),

University of Patras, GR-26500 Patras, Greece

3Observatoire de Marseille, 2 Place Le Verrier, F-13248 Marseille Cedex 4, France

Accepted . Received ; in original form

ABSTRACT

We study the conditions that favour boxiness of isodensities in the face-on views of

orbital 3D models for barred galaxies. Using orbital weighted profiles we show that

boxiness is in general a composite effect that appears when one considers stable orbits

belonging to several families of periodic orbits. 3D orbits that are introduced due

to vertical instabilities, play a crucial role in the face-on profiles and enhance their

rectangularity. This happens because at the 4:1 radial resonance region we have several

orbits with boxy face-on projections, instead of few rectangular-like x1 orbits, which,

in a fair fraction of the models studied so far, are unstable at this region. Massive bars

are characterized by rectangular-like orbits. However, we find that it is the pattern

speed that affects most the elongation of the boxy feature, in the sense that fast bars

are more elongated than slow ones. Boxiness in intermediate distances between the

center of the model and the end of the bar can be attributed to x1v1 orbits, or to a

combination of families related to the radial 3:1 resonance.

Key words: Galaxies: evolution – kinematics and dynamics – structure

1INTRODUCTION

Boxy isophotes are a typical feature at the end of the bars of

early type (SB0, SBa) barred galaxies seen not far from face-

on. Typical examples can be found in Athanassoula, Morin,

Wozniak et al. (1990) (NGC 936, NGC 4314, NGC 4596), in

Buta (1986) (NGC 1433), in Ohta, Hamabe and Wakamatsu

(1990) (NGC 2217, NGC 4440, NGC 4643, NGC 4665)

and in many other papers. Loosely speaking, the shape of

these isophotes is rectangular-like. Their main characteristic

is that their small sides, at the largest distance of the

isophotes from the galactic center, are roughly parallel to

the bar minor axis (Athanassoula 1984, Athanassoula et

al. 1990, Elmegreen 1996). In particular, Athanassoula et

al. (1990) use generalized ellipses to fit the isophotes of

the bars in a sample of early-type strongly barred galaxies,

thus describing quantitatively the result that the shapes of

the isophotes are rectangular-like rather than elliptical-like.

Fig. 1 is a DSS image of NGC 4314 in B and demonstrates

a typical case of a galaxy with boxy isophotes at the end

of its bar. We observe that the last isophotes of the bar

are indeed rectangular-like. Beyond the area of the boxy

isophotes starts the spiral structure of this galaxy.

There is a correspondence between the morphology

of the boxy isophotes of the early type barred galaxies

and the isodensities encountered in snapshots of several

Figure 1. DSS image of NGC 4314 in B.

N-body models of bars. This was shown for the first

time in a simulation that was run specifically for this

purpose (Athanassoula et al. 1990, Fig. 7). In that paper

this correspondence was underlined by measuring the

rectangularity in the same way as in the observations.

Since then there have been several snapshots in N-body

simulations reproducing this morphological feature (see e.g.

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2P.A. Patsis et al.

Shaw, Combes, Axon et al. 1993; Friedli & Benz 1993;

Debattista & Sellwood 2000). Recently Athanassoula &

Misiriotis (2002) in their N-body models describe this

feature also quantitatively. We can thus conclude that

both observations and numerical models clearly show that

boxiness close to the end of the bars is a very frequently

encountered phenomenon, and thus it should be related to

the standard dynamical behaviour in such systems.

Early calculations of orbits in 2D static potentials have

underlined the presence of rectangular-like periodic orbits at

the 4:1 resonance region (Athanassoula, Bienayme, Martinet

et al. 1983; Contopoulos 1988; Athanassoula 1992a). They

are either orbits of the x1 family on the decreasing part

of the characteristic – towards lower x values – in type-2

4:1 resonance gaps⋆, or orbits at the ‘4:1 branch’ beyond

the type-1 gap (Contopoulos 1988). These orbits, whenever

they exist, support outer boxiness on the face-on views of

the models. Their mere presence, however, is not sufficient

to explain the observed morphology. Explanations based on

the presence of rectangular-like planar 2D x1 orbits suffer

from the following problems:

• The range of the Jacobi integral†over which one finds

stable rectangular-like x1 orbits (in type-2 gaps) or 4:1

orbits (in type-1 gaps) is in general narrow (Contopoulos

and Grosbøl 1989). Furthermore, these orbits develop loops

at the four corners, whose size increases considerably with

energy, while the near-horizontal sections of the orbit

approach the minor axis (as in Fig. 3d in Athanassoula

1992a). This happens for energies only a little larger than the

energy at which the orbits become rectangular-like. Orbits

with loops cannot easily reproduce the observed boxiness.

Thus, we have only a very small energy interval with useful

orbits (see Fig. 8 in Athanassoula et al. 1990).

• Poincar´ e sections for an energy value within the small

energy interval where rectangular-like orbits are stable,

show that the size of the stability area is very small

(see e.g. Fig. 21 in Patsis et al. 1997a). This renders the

trapping of many non-periodic orbits around stable boxy

periodic orbits rather difficult. It is characteristic that

Patsis, Efthymiopoulos, Contopoulos et al. (1997b) used

dynamical spectra in order to trace tiny stability islands

of rectangular-like orbits in a 2D Ferrers bar potential.

• Inmanymodelsthe orbital

sufficiently elongated. They frequently are rather square-

like, even in cases of strong bars (as in Fig. 3c in

Athanassoula 1992a).

rectanglesarenot

Besides the papers of Contopoulos on non-linear

phenomena at the 4:1 resonance region in 2D bars in the

’80s, and the work of Athanassoula and collaborators on

the morphology of bar orbits in the early ’90s, where the

problem is explicitely stated, not much work has been done

on this issue. Nevertheless, from figures of orbits in models of

3D bars (Pfenniger 1984; 1985) one can infer that problems

like the squareness of the rectangular-like orbits persist even

if one considers also the third dimension of the bars. The

⋆for the nomenclature of the gaps see Contopoulos (1988), or

Contopoulos & Grosbøl (1989).

†We will hereafter refer to the Jacobi integral as the ‘energy’.

morphology of single orbits, however, does not determine

the boxiness of the isodensities of a model.

The goal of the present paper is to reconsider the

problem of rectangular-like isodensities in the framework

of 3D orbital structure models, and, more specifically, to

examine the contribution of the third dimension to the

face-on orbital profiles. Based on the models presented in

Skokos, Patsis & Athanassoula (2002a,b; Papers I and II

respectively), we investigate the parameters which favour

the presence of boxy outer isophotes in the face-on profiles of

3D bars. As in most orbital structure studies, the models we

present are not self-consistent. As underlined in Papers I and

II, our goal is to study the orbital behaviour, and therefore

the morphological changes, as a function of the model

parameters. For this we consider even extreme cases, in order

to make the effects clearest. We thus determine the main

parameters that influence the boxiness of a model. We base

our results not on the morphology of single orbits, but on

collective appearance when orbits of more than one family

are taken into account simultaneously. In parallel to this we

study, as counterexamples, cases where the appearance of

rectangular-like orbits close to the end of the bar is excluded.

Finally, we discuss the correspondence between boxiness

observed in edge-on profiles and boxiness observed in the

middle of the bars when viewed face-on.

2FACE-ON PROFILES

In this series of papers (Papers I, II, and Patsis, Skokos &

Athanassoula 2002, hereafter Paper III) we study the basic

families in a general model composed of a Miyamoto disc

of length scales A=3 and B=1, a Plummer sphere bulge of

scale length 0.4 and a Ferrers bar of index 2 and axial ratio

a : b : c = 6 : 1.5 : 0.6. The masses of the three components

satisfy G(MD+MS+MB) = 1 and are given in Table 1. The

length unit is 1 kpc, the time unit is 1 Myr and the mass

unit is 2 × 1011M⊙. In the present paper we examine two

additional models. One of them, model B2, is characterized

by a very strong bar, whose mass is 40% of the total, and

whose remaining parameters are as in model B (Paper II).

The other additional model has different axial ratios than

the rest of our model bars. It has a : b : c = 6 : 1 : 0.6,

instead of a : b : c = 6 : 1.5 : 0.6, as all others. It is used to

study the contribution of families related to the radial 3:1

resonance to the appearance of boxy isophotes at the end

of bars. The rest of its parameters are as in model A1, and

we thus call it A1b. The basic properties of the models we

present in this paper are summarized in Table 1.

A fundamental conclusion of Papers I and II is that

essentially the backbone for building 3D bars is the x1 tree of

families of periodic orbits. The tree consists of the x1 planar

orbits and of its 3D bifurcations at the vertical resonances.

Only in one case (family z3.1s in model B) did we find a

family which supports the bar without being introduced

in the system after a x1 bifurcation. In the present paper,

we use these families in order to present the face-on views

of the skeletons of the models. We use for this purpose

sets of weighted orbits as in Paper III. As we explained

in Paper III, in order to build a profile of weighted orbits

for a model, we first calculate a set of periodic orbits and

pick points along each orbit at equal time steps. We keep

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Orbital dynamics of three-dimensional bars:IV. Boxy isophotes in face-on views3

only stable representatives of a family. The ‘mean density’

of each orbit (see §2.2 of paper III) is considered as a

first approximation of the orbit’s importance and is used

to weight the orbit. We construct an image (normalized by

its total intensity) for each calculated and weighted orbit,

and then, by combining sets of such orbits, we construct

a weighted profile. The selected stable orbits are equally

spaced in their mean radius. The step in mean radius is

the same for all families in a model. We underline the fact

that the orbital profiles we present throughout the paper

comprise only stable periodic orbits.

As we have seen, the edge-on orbital profiles (paper III)

are of stair-type, which means that the families building the

outer parts of the bars have the lowest |z|. We are thus here

mainly interested in orbits remaining close to the equatorial

plane, since these orbits will contribute more to the surface

density at the end of the bars.

2.1Model A1

The fiducial case model A1 offers, also for the face-

on structure of the models, a typical example of the

contribution of the individual families to the observed face-

on orbital morphology. Fig. 2 shows the weighted profiles of

all contributing families. It is evident by simple inspection

that the orbits of the 2D family x1 (Fig. 2a) are the most

important, mainly because they have stable representatives

over a large energy range. Nevertheless, the projections of

the 3D families depicted from Fig. 2c to Fig. 2h play an

important role. Contrarily, the x2 orbits (Fig. 2b) affect

only the central parts of the system. As we can see, boxy

features are related to the families x1v1 (Fig. 2c), x1v4

(Fig. 2e), x1v5 (Fig. 2f), x1v7 (Fig. 2g) and x1v9 (Fig. 2h).

The family x1v3 (Fig. 2d), on the other hand, has always

orbits with elliptical-like projections, thus in a way plays

a complementary role, together with x1, providing building

blocks for the elliptical-like part of the bar.

The orbits with the rectangular-like projections in

model A1 face one of the main problems for explaining the

boxy isophotes, i.e. they are less elongated than necessary.

In general the boxy isophotes of the early type bars are

not squares. The rectangular-like orbits in model A1 have

also shorter projections than the x1 ellipses on the semi-

major axis. An exception is x1v9 (Fig. 2h). However, the

four loops of these orbits are larger than can be admitted

by the corresponding shapes of the isophotes of real galaxies

Furthermore, the x1v9 family contributes, because of its

stability, only over a narrow energy interval (−0.185 < Ej <

−0.182).

It is interesting to note that the boxy x1v1 orbits,

responsible in many models for peanut-shaped edge-on

profiles, provide to the face-on view of the system a ‘bow

tie’ structure. However, in this particular case at least, these

orbits remain confined well inside corotation and can be

responsible only for an inner boxiness in a galaxy and not

for boxy isophotes at the end of the bars.

Fig. 3 combines all x1 related families, i.e. the 2D x1

family and its 3D bifurcations, as well as the x2 and the long-

period banana-like orbits. We can see that the orbits support

a bar with a semi-major axis of length about 0.75 of the

corotation radius, so that the ratio of the corotation radius

Rc to the orbital length of the bar ao will be Rc/ao = 1.33.

Figure 3. Face-on orbital profile for model A1. All x1-related

orbits and the banana-like orbits are included, and they are

weighted as described in paper III.

The longest orbits along the bar are elliptical-like. This

however, is not an obstacle to forming more rectangular-

like bars, since the elliptical-like orbits extend only little

beyond the rectangular-like ones. This extra extent could

be suppressed if the outermost periodic elliptical-like orbits

were not populated, or it could be masked by orbits trapped

around the periodic rectangular-like ones. In the latter

case, the isophotes (or rather isodensities) would be more

elongated than the corresponding rectangular-like orbits.

We note that in a galaxy or in an N-body simulation,

not all families included in the figure should be necessarily

populated. In our non-self-consistent models we try to

identify structures in order to seek their corresponding

features in galactic images and snapshots of self-gravitating

models.

2.2Model A2

The slow rotating bar in model A2 brings new morphological

features. In this case square- or rectangular-like orbits play

a minor role, as we realize from Fig. 4. Only x1′orbits

(Fig. 4b) at the decreasing branch of the characteristic

(paper II) and part of the x1v1 family, after the S→∆

transition (Fig. 4c), provide such stable orbits to the system.

For both these families, however, the size of the orbits and

the energy range over which they exist do not allow them to

play a major role in the orbital structure of the model. In

Fig. 4b we choose the contrast of the image such as to allow

us to see the loops of the x1′square-like orbits. For even

larger energies the loops of the x1′square orbits become

huge, and finally the orbits become retrograde. Such orbits

are not depicted in Fig. 4b.

Model A2 has two main features: First, the loops

along the bar major axis, which are brought in the system

by the x1 family and its x1v3 and x1v4 bifurcations.

Second, the almost circular (and/or square-like) projections

on the equatorial plane of the families x1′, x1′v4, and

x1′v5 (Fig. 4b, f, h respectively). The face-on morphology

supported by this slow rotating bar is depicted in Fig. 5. In

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4P.A. Patsis et al.

Table 1. Parameters of our models. G is the gravitational constant, MD, MB, MSare the masses of the disk, the bar and the bulge

respectively, ǫs is the scale length of the bulge, Ωbis the pattern speed of the bar, Rc is the corotation radius. The comment in the last

column characterizes the model in order to facilitate its identification.

model name GMD

GMB

GMS

ǫs

Ωb

Rc

comments

A1

A2

A3

B

C

D

B2

A1b

0.82

0.82

0.82

0.90

0.82

0.72

0.60

0.82

0.1

0.1

0.1

0.1

0.1

0.2

0.4

0.1

0.08

0.08

0.08

0.00

0.08

0.08

0.00

0.08

0.4

0.4

0.4

–

1.0

0.4

–

0.4

0.0540

0.0200

0.0837

0.0540

0.0540

0.0540

0.0540

0.0540

6.13

13.24

4.19

6.00

6.12

6.31

6.51

6.10

fiducial

slow bar

fast bar

no bulge

extended bulge

strong bar

no bulge/very strong bar

a : b : c = 6 : 1 : 0.6

Figure 2. The face-on, (x,y), weighted profiles of the 3D families in model A1. (a) x1, (b) x2, (c) x1v1, (d) x1v3, (e) x1v4, (f) x1v5,

(g) x1v7, and (h) x1v9.

this figure we combine all orbits of the families presented

in Fig. 4. The result is a bar with loops along the major

axis surrounded by almost circular orbits. We note that the

bar-supporting orbits extend to a distance about 9 from the

center (corotation in this case is at 13.24), i.e. in this case

Rc/ao = 1.5.

2.3 Model A3

If the bar rotates fast (model A3), the face-on orbital

structure changes significantly. The x1 family dominates

once again, as we can see in Fig. 6a. Its orbits remain

always elliptical-like, but now they support a bar with

length 0.95 of the corotation radius, i.e. Rc/ao = 1.05. The

dynamics at the radial 4:1 resonance region are crucial for

the morphology of the model (paper II). In this case the

families q0 and x1v8 provide the system with rectangular-

like orbits quite elongated along the major axis of the

bar. Family q0 can be found in two branches, symmetric

with respect to the bar’s major axis. We can, however,

consider orbits of only one of its two branches if we want

in our weighted profiles a non-rectangular parallelogram-

like morphology. In order to obtain a desired morphology

in our response models, we chose any combination of stable

orbits from all available families. In the present case both

families q0 and x1v8 have orbits at least as long as the x1

orbits. In addition, families x1v1 and x1v5 also have orbits

with boxy projections on the equatorial plane, but reaching

distances from the center 1.2 and 3.2 respectively (Fig. 6c,e).

That means that, if all boxy families are populated, we can

have boxy isophotes in several scales in this model. Even

the hexagonal-like projections of the orbits of family x1v3

(Fig. 6d) contribute to the boxiness of the model with their

sides which are parallel to the bar major axis. The face-on

view of this family has also the ‘bow tie’ appearance.

The total effect when considering all orbits is given in

Fig. 7, where we take into account all families of Fig. 6.

In Fig. 7a we give all weighted orbits together, while in

Fig. 7b we apply a gaussian filter in order to show clearly, in

a first approximation, the shapes of features, which could be

supported by the orbits in the density maps of the models.

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Orbital dynamics of three-dimensional bars:IV. Boxy isophotes in face-on views5

Figure 4. The face-on, (x,y), profiles of the 3D families in model A2. (a) x1 and x2, (b) x1′, (c) x1v1, (d) x1v3, (e) x1v4, (f) x1′v4, and

(h) x1′v5.

b

d

c

f

a

e

Figure 6. The face-on, (x,y), profiles of the 3D families in model A3. (a) x1, (b) q0, (c) x1v1, (d) x1v3, (e) x1v5, (f) x1v8.

Fig. 7 tells us that the fast rotating bar model, has a boxy

bar with Rc/ao ≈ 1.05.

The fast rotating bar case offers the opportunity to

study models with a substantial non-axisymmetric force

near corotation. In this way we compensate for the standard

shortcoming of Ferrer’s‡ellipsoids, namely that their force

‡We use Ferrers bars because we are not aware of any other

drops too fast as the radius increases and approaches

corotation. Thus A3 allows us to cover also cases with

substantial non-axisymmetric forcing near corotation.

models which are realistic and analytic, and do not have this

shortcoming.

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6P.A. Patsis et al.

a

b

d

c

Figure 8. The face-on, (x,y), profiles of the 3D families in model B. (a) x1, (b) x1v5 and x1v5′, (c) x1v7, (d) z3.1s

Figure 5. Face-on orbital profile for model A2. All orbits are

weighted as described in paper III.

2.4 Model B

Model B is a model without radial or vertical 2:1 resonances.

As we have seen in paper II, the first vertical bifurcation of

x1 is x1v5. Thus, the families that build the bar are x1,

x1v5/x1v5′, x1v7 and the family z3.1s. We remind that this

latter family plays a significant role in the orbital behaviour

of this particular model (see Paper II) and it was found as a

bifurcation of the z-axis orbits when the latter are considered

as being of multiplicity 3, i.e. the orbits are repeated three

times. The z3.1s family is not related to the x1-tree. The

weighted face-on profiles of the above mentioned families

are given in Fig. 8. We see that all of them have boxy

representatives in their projections on the equatorial plane.

In Fig. 9 we have the orbital face-on view for model B

obtained by overplotting the weighted orbits of all families

together. In this model Rc/ao ≈ 1.4.

2.5 Model D

The strong bar model D is, after the fast rotating bar model

A3, the second case where we have periodic orbits that

a

b

Figure 7. (a) Composite profiles combining all orbits in model

A3. (b) The profile after applying a gaussian filter in order to

get an impression of the morphological features supported by the

orbits.

can contribute to sufficiently elongated rectangular-like boxy

isophotes. They are, however, considerably less elongated

than the rectangular orbits in A3. As we see in Fig. 10, the

main contributors are now families x1, x1v5/x1v5′and x1v7.

Again x1v1 orbits have ‘bow tie’ face-on projections, and

support a feature of corresponding morphology (Fig. 10b).

This feature occupies the main part of the bar, but it does

not reach its end, which is at about a distance 0.9 of the

corotation radius.

The composite face-on orbital profile of model D is

given in Fig. 11. We see that –although the boxy orbits are

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Orbital dynamics of three-dimensional bars:IV. Boxy isophotes in face-on views7

a

b

de

c

f

Figure 10. The face-on, (x,y), profiles of the 3D families in model D. (a) x1, (b) x1v1, (c) x1v3, (d) x1v5/x1v5′, (e) x1v6, (f) x1v7

Figure 9. Composite profile combining the orbits of families x1,

x1v5/x1v5′, x1v7 and z3.1s in model B.

present and, if populated, could characterize the morphology

of the model– the dominant feature is the loops of the x1

orbits along the major axis. This morphology is enhanced

by the loops of the projections of the 3D bifurcations of x1,

x1v5/x1v5′and x1v6 (Fig. 10). It should be noted, however,

that only the loops of x1v6 and some of the x1 loops extend

beyond the rectangular outline. If we omit orbits with such

loops we get, for this model, Rc/ao ≈ 1.23, while if we

include them we get Rc/ao ≈ 1.08.

Figure 11. Composite face-on profile of model D. All orbits of

Fig. 10 are considered.

2.6Very massive bars

In all the models we studied so far the rectangular-like

orbits, if they exist, are rather square-like, except for the

fast rotating bar case (model A3), where they are more

elongated. The second parameter, after the pattern speed,

which made the rectangular-like orbits more elongated is

the increase of the relative mass of the bar. For this

reason, we studied several more models with increased bar

mass fraction and calculated the orbital stability of each

individual case. We stopped when we reached models with

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8 P.A. Patsis et al.

a

b

Figure 12. (a) A composite face-on profile of model B2. We consider orbits from the families x1, x1v5/x1v5′and x1v7. (b) The blurred

profile of model B2. Besides the x1, x1v5/x1v5′and x1v7 we consider stable orbits of x1v6 and 2D orbits of family t1. In model B2 we

can have rectangular-like face-on profiles with pmax/pminratio close to 2.

very large intervals of instability of the x1 family and its 3D

bifurcations. From a sequence of models starting with the

values of the parameters of model B and with an increasingly

large fraction of mass in the bar, we found that this fraction

is about 50% of the total mass. It was evident that models

with bar masses about 40% of the total mass offered families

of periodic orbits with enough stable representatives to

support sufficiently elongated rectangular-like isophotes at

the end of the bars. E.g. model B2, which differs from model

B only in that the mass of the bar GMB=0.4 instead of 0.1

as in model B, has rectangular-like orbits with the ratio

of their projections on the major axis of the bar to their

projections on the minor axis (pmax/pmin) close to 2. This

can be seen in Fig. 12. In Fig. 12a we have orbits from the

families x1, x1v5/x1v5′and x1v7, omitting rectangular-like

orbits with loops. The blurred image in Fig. 12b includes

also orbits of the t1 family (see Paper I) and stable orbits

of family x1v6. The latter is initially bifurcated as unstable.

For larger energies, however, it has stable orbits with face-on

projections with loops along the major axis of the bar. The

t1 orbits, bifurcated at the radial 3:1 resonance, introduce

in the system some interesting features if one considers both

their branches which are symmetric with respect to the

minor axis of the bar at a given energy. The t1 orbits are

responsible for the guitar-like feature we observe in Fig. 12b

closer to the major axis than the outer rectangular-like

orbits. For model B2 we have Rc/ao ≈ 1.04.

2.7Support by 3:1 families

The contribution of t1 to the features we found in model

B2 gave us the incentive to investigate the contribution of

families bifurcated at the radial 3:1 resonance in all our

models. We found that in general orbits from the t1 family

may support motion roughly parallel to the minor axis of

the bar, but not at the end of the bar. Fig. 12b describes

the kind of contribution t1 orbits can offer to the overall

Figure 13. x1 and t2 orbits in model A1b. The elliptical-like

orbits with loops along the major axis of the bar are clearly

separated from the rest of the orbits. Due to orbital instabilities

the boundary between the two regions has a rectangular-like

shape and is enhanced by the t2 orbits. Its four corners are

indicated by arrows.

boxiness of face-on profiles of barred galaxies. We give also

an example of the combination of t2 orbits (see Paper I) in a

model in which this contribution is pronounced. It is model

A1b, a model that differs from the fiducial case (model A1,

Paper I) only in the axial ratios, which in model A1b are

a : b : c = 6 : 1 : 0.6 instead of a : b : c = 6 : 1.5 : 0.6.

In Fig. 13 we observe stable x1 orbits with loops along the

major axis of the bar, and x1 orbits with a more or less

rectangular like shape. The two groups are separated by

an empty region caused by an instability region of the x1

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Orbital dynamics of three-dimensional bars:IV. Boxy isophotes in face-on views9

Table 2. The ratio of the corotation radius Rc to the orbital

length of the bar ao (Rc/ao) for all models studied in the present

paper. We find 1.04<

=Rc/ao<

=1.5. The third column gives a

short comment about what characterizes the model in order to

facilitate its identification.

model nameRc/ao

comments

A1

A2

A3

B

D

B2

A1b

1.33

1.50

1.05

1.40

1.08

1.04

1.35

fiducial

slow bar

fast bar

no bulge

strong bar

no bulge/very strong bar

a : b : c = 6 : 1 : 0.6

family. On top of the x1 orbits we overplot stable orbits of

both branches, symmetric with respect to the major axis

of the bar, of the t2 family. The t2 orbits contribute to the

boxiness of the model by enhancing the borders of the empty

region and by forming a rectangular-like feature indicated in

Fig. 13 by four black arrows. Model A1b has Rc/ao ≈ 1.35.

3DISCUSSION

We examined the face-on views of all models of Papers I

and II and we found families that could support outer,

as well as inner, boxy isophotes. In the present paper

we do not refer explicitly to model C, since its orbital

behaviour does not differ essentially from that of model A1.

Furthermore, we use composite profiles of two additional

models to demonstrate the effect of very massive barred

components and the possible role of families related to the

radial 3:1 resonance.

The first quantity that can be used to compare the

bars of our models with the bars of real galaxies concerns

the length of the bar built by the orbits with respect to

corotation radius. Indeed observations (for a compilation

see e.g. Athanassoula 1992b, Elmegreen 1996 and Gerssen

2002) as well as hydrodynamical simulations (Athanassoula

1992b), have shown that the ratio Rc/ao takes values only

in a restricted range, namely Rc/ao = 1.2 ± 0.2. This

is in agreement with 2D orbital models by Contopoulos

(Contopoulos 1980). All our response models have Rc/ao

ratios within this range, except for model A2 which has a

slow rotating bar and Rc/ao = 1.5. Table 2 summarizes the

Rc/ao ratios for the models we studied in the present paper.

A second quantity we can compare with galaxies and

snapshots of N-body models is the ratio pmax/pmin for the

rectangular-like orbits. Eye estimates show that the boxy

outer isophotes at the end of the bar of strongly barred

galaxies have a ratio pmax/pmin typically larger than 2

(e.g. NGC 936, NGC 4314, NGC 4596). Exceptional cases

can be found in the literature but are not many (see e.g.

the not-rectified images of NGC 1415 in Garcia-Barreto &

Moreno (2000)). A more quantitative study has been made

by Athanassoula et al (1990). They find that the axial ratios

of the isophotes near the end of the bar are, for all the

galaxies in their early-type strongly-barred galaxy sample,

considerably larger than 2. On the other hand, in most of

our models the boxy orbits are quite square-like. Ratios of

boxy isophotes larger than 2 have been mainly found in the

Figure 14. Isodensities on a blurred image of model B. The

narrow rectangular-like structure is supported by orbits of more

than one family.

fast rotating bar case, where pmax/pmin ≈ 2.5. The strong

bar model D has pmax/pmin ≈ 1.4, while in model B2, where

the mass of the bar is 40% of the total mass, we attained

a pmax/pmin ratio close to 2. In the rest of the models the

ratio was less than 1.4.

Let us underline that, for studying the elongation of

the rectangular-like isophotes observed in barred galaxies,

one needs to combine orbits of several families instead of

evaluating the properties of a single orbit or a single family.

This is a result of the three dimensional character of our

models and of the fact that we have bars built by orbits

belonging mainly to the x1-tree. This effect is indicated

by isodensities we plotted on some blurred images of our

models. Such a typical case is given in Fig. 14, for model B.

The small sides of the rectangular-like structure outlined

by the isophotes reflect mainly the contribution of the

rectangular-like orbits. However, the two large sides, parallel

to the major axis of the bar, are due to the overlapping of

orbits of the x1, and z3.1s families.

Our models show that it is the pattern speed that

mainly determines the elongation of the outer boxy orbits,.

The orbits building the bar of model A3 are rectangular-

like and could make a bar with the geometry encountered in

the early-type bars with outer boxy isophotes. It is just the

increase of Ωb that brought in the system the 2D family q0

and the 3D, stable x1v8 orbits. We note the morphological

similarity of the q0 orbits with the outer boxy isophotes of

NGC 4314 (Quillen, Frogel & Gonzalez 1994), which have

the form of a non-rectangular parallelogram (cf. Fig. 7b with

Fig. 1). We note that in model A3 the rectangular-like bar

ends closer to corotation than the bars of the models for

which the longest orbits are the x1 with loops along the

major axis. This is consistent with the result found in Patsis

et al. (1997a) for the pattern speed of NGC 4314, in which

the boxy isophotes are very close to the corotation radius

indeed. It is also obvious that, apart from the increase of

the pattern speed, the increase of the strength of the bar

favoured the elongation of the rectangular-like orbits.

Model A2, which is the slowest rotating case, is the

other extreme of the face-on morphologies we encounter

c ? 2001 RAS, MNRAS 000, 1–11

Page 10

10P.A. Patsis et al.

in our models. In this model, planar boxy orbits and

projections of orbits on the equatorial plane are square-like,

and very little elongated along the major axis of the bar

(Fig. 4c of the present paper, and Fig.5 in paper II). The

development of loops at the apocentra of the x1 elliptical-

like orbits, as well as in the projections on the equatorial

plane of the x1v3 and x1v4 families, together with the

almost circular x1′orbits and the projections of the x1′v4

and x1′v5 families, give to model A2 a characteristic face-

on morphology with the circular-like orbits surrounding all

other orbits in the face-on projection. Such a morphology

could be linked to the existence of inner rings in barred

galaxies.

Inner boxiness of the face-on profiles, much closer to

the center than the corotation region, is associated mainly

with the x1v1 orbits, i.e. with the 3D family born at the

vertical 2:1 resonance. As we have seen in paper III, in

several models this family is responsible for peanut-shaped

orbital structures in the edge-on views of the models. The

face-on orbital skeletons of the models we present here show

in another clear way what we noticed in paper III, namely

that the x1v1 family builds boxy- or peanut-shaped features

which do not approach corotation. Inner boxiness is not rare

in galactic bars. Typical examples of boxy isophotes in the

middle of the stellar bars are NGC 3992 and NGC 7479

(Wilke, M¨ ollenhoff & Matthias 2000). The x1v1 orbits, or at

least their representatives with the largest energy values (see

Fig. 2c, Fig. 6c and Fig. 10b), are boxy in their face-on views,

but they also have a characteristic ‘bow-tie’ morphology.

Inner boxiness is also supported by the z3.1s orbits, at large

values of the energy, in model B.

A ‘bow-tie’ morphology in models of the kind we study

here can also be introduced by rectangular-like orbits at

high energies. These orbits (or more precisely their face-

on projections) develop loops, which, in some cases, stay

close to the orbits that build the bulk of the bar. This

is e.g. the case of the x1v5′family in model B (Fig. 8b).

When the rectangular-like orbits develop loops they also

become of ‘bow-tie’ shape. Since their deviations from

the equatorial plane increase with energy, their projections

occupy almost the same area as the rectangular-like orbits

of the same family, for lower energies, do. In model B, for

EJ ≈ −0.185, integrating even chaotic orbits for a small

number of dynamical times (of the order of 30) we get a

bow tie morphology. This is shown in Fig. 15. Non-periodic

elliptical-like orbits trapped around the x1 family not far

from the center of the bar, as well as the projection of orbits

trapped around the three-dimensional x1 bifurcations would

fill more densely the area between the regions A and B (and

symmetrically between C and D), than between the regions

A and C (and B and D). This results from the elliptical-like

shape of the projections of these orbits on the equatorial

plane and the fact that they have their apocentra between

the regions A and B (and C and D). We note also the

contribution of the x1v1 orbits to a ‘bow-tie’ morphology

of face-on profiles, as mentioned previously.

Due to the presence of these families a bow-tie

morphology, like the one found by Barnes & Tohline

(2001), is not excluded from the usual barred morphology.

Nevertheless, we did not find orbits of a single family that

can both enhance a rectangular-like structure at the end of

the bar without loops at its four corners, and simultaneously

A

B

C

D

Figure 15. An example of a chaotic orbit in Model B, that,

for a small number of dynamical times, supports a ‘bow-tie’

morphology. A particle following this trajectory spends most of

the time in the regions of the loops indicated with A, B, C and D,

⌢

AC andBD approach the center of the galaxy.

Darker areas indicate regions where the orbit spends more time.

while the ‘arcs’

⌢

enhance a bow-tie morphology. Single families with bow-tie

profiles in their face-on projections do not extend to the end

of the bar (x1v1 in Figs. 2c, 6c, 10b and x1v3 in Fig. 6d).

4CONCLUSIONS

In this paper we discussed boxiness in the face-on views

of 3D models, and, in general, their face-on orbital

structure. We examined a large variety of possible 3D orbital

behaviour, that could contribute to the boxiness in face-on

views of barred galaxies. Our main conclusions are:

(i) Boxiness in the face-on views of 3D barred models is

an effect caused by the co-existence of several families, each

contributing appropriate stable orbits. The morphology of

boxy isodensities/isophotes is not necessarily similar to the

morphology of individual stable, rectangular-like orbits. In

some cases (Fig. 14) the iso-contours can be narrower than

the orbits.

(ii) In models with the right morphological parameters,

we find appropriate building blocks to account for the

rectangular-like isophotes or isodensities seen in early type

barred galaxies and in some N-body simulations.

(iii) In 3D models the family of the planar x1 orbits is

subject to vertical instabilities, and thus in several cases

it has considerable instability strips at the 4:1 resonance

region. This should be an obstacle for x1 planar orbits to

account for the rectangular-like shape of bars, since unstable

periodic orbits can not trap regular orbits around them.

However, at the instability regions of the x1 we find other

stable families, whose (x,y)-projected orbital shapes are, at

least near their bifurcations, very similar to those of the

x1. They have orbits which are stable over large energy

intervals and also have (x,y)-projected shapes that can

enhance a rectangular-like bar outline. Thus the inclusion of

c ? 2001 RAS, MNRAS 000, 1–11

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Orbital dynamics of three-dimensional bars:IV. Boxy isophotes in face-on views11

the third dimension in the models enhances the possibility

of rectangular-like isodensities.

(iv) There are families of periodic orbits that support

boxiness in the outer bar regions, as well as families

that support boxiness in somewhat more inner parts. The

standard families belonging to the former category are the

stable representatives of the x1 orbits (close to the radial

4:1 resonance), and of the families x1v5, x1v7 and x1v9.

The families q0 and x1v8 play a major role in model A3 and

thus could be essential for fast rotating bars. Nevertheless,

the geometry of a boxy feature becomes evident mainly in

weighted profiles, where orbits from one or more families

are considered. Inner boxiness is associated mainly with the

x1v1 family and, in model B, with family z3.1s.

(v) Orbits of the families related to the 3:1 resonance (t1

and t2) may contribute in some cases to the boxiness of

the profiles. The t1 family can do this by supporting motion

parallel to the minor axis of the bar in intermediate distances

between the galaxy center and corotation, and the t2 family

by enhancing the sides of the boxes parallel to the major

axis.

(vi) The consideration of several families of orbits for

building a profile may lead to boxy features close to the

end of bars, with pmax/pmin ratios different from the

corresponding ratios of individual orbits or families of orbits.

(vii) An essential conclusion of our investigation is that

outer boxiness is favoured by fast bar pattern speeds, while

in the slow-rotating model the bar is surrounded by almost

circular orbits. These are the two extremes of an orbital

behaviour that changes as the pattern speed varies from

one model to the other. The near-circular orbits could be

building blocks for inner rings.

(viii) The fast rotating bar has a length 0.95 of its

corotation radius, while the slow one only 0.68. This

indicates that boxy bars end close to their corotation, while

the end in slow rotating bars might be associated with n : 1

resonances of lower n values.

(ix) The rectangular-like orbits in models with faster bars

are more elongated than the corresponding orbits in models

with slower bars. When modeling individual barred galaxies,

for which we know a priori that in general 1 < Rc/ao < 1.4,

the elongation of the orbits can be used to give estimates of,

or at least set limits to, the bar pattern speed, without the

use of kinematics, which is not always available.

(x) The second mostefficient

rectangular-like orbits is to increase the mass of the bar. This

mechanism, however, is restricted by the fact that in very

strong bars the families of periodic orbits which support the

boxy face-on profiles are unstable over large energy intervals.

(xi) Outer boxy, ‘bow tie’ morphology is possible in some

models by combining orbits of several families. A necessary

condition for this is to have 4:1 type orbits with loops close

to the main bar. Inner ‘bow tie’ morphology can be due to

the presence of x1v1 orbits (Fig. 2c, Fig. 6c, Fig. 10b).

way ofstretching

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ACKNOWLEDGMENTS

We

comments by G. Contopoulos and A. Bosma. We thank

the anonymous referee for valuable remarks, which improved

the paper. This work has been supported by EΠET II and

KΠΣ 1994-1999; and by the Research Committee of the

Academy of Athens. ChS and PAP thank the Laboratoire

d’Astrophysique de Marseille, for an invitation during which,

essential parts of this work have been completed. ChS was

partially supported by the “Karatheodory” post-doctoral

fellowship No 2794 of the University of Patras. All image

processing work has been done with ESO-MIDAS. DSS was

produced at STSI under U.S. Government grant NAG W-

2166. The images of these surveys are based on photographic

data obtained using the Oschin Schmidt Telescope on

Palomar Mountain and the UK Schmidt Telescope.

acknowledgefruitfuldiscussionsand veryuseful

This paper has been produced using the Royal Astronomical

Society/Blackwell Science LATEX style file.

c ? 2001 RAS, MNRAS 000, 1–11