Orbital dynamics of three-dimensional bars: IV. Boxy isophotes in face-on views
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.
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ABSTRACT: We study the distinction and quantification of chaotic and regular motion in a time-dependent Hamiltonian barred galaxy model. Recently, a strong correlation was found between the strength of the bar and the presence of chaotic motion in this system, as models with relatively strong bars were shown to exhibit stronger chaotic behavior compared to those having a weaker bar component. Here, we attempt to further explore this connection by studying the interplay between chaotic and regular behavior of star orbits when the parameters of the model evolve in time. This happens for example when one introduces linear time dependence in the mass parameters of the model to mimic, in some general sense, the effect of self-consistent interactions of the actual N-body problem. We thus observe, in this simple time-dependent model also, that the increase of the bar's mass leads to an increase of the system's chaoticity. We propose a new way of using the Generalized Alignment Index (GALI) method as a reliable criterion to estimate the relative fraction of chaotic vs. regular orbits in such time-dependent potentials, which proves to be much more efficient than the computation of Lyapunov exponents. In particular, GALI is able to capture subtle changes in the nature of an orbit (or ensemble of orbits) even for relatively small time intervals, which makes it ideal for detecting dynamical transitions in time-dependent systems.Journal of Physics A Mathematical and Theoretical 06/2013; 46:254017. · 1.77 Impact Factor
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ABSTRACT: Bars play a major role in driving the evolution of disk galaxies and in shaping their present properties. They cause angular momentum to be redistributed within the galaxy, emitted mainly from (near-)resonant material at the inner Lindblad resonance of the bar, and absorbed mainly by (near-)resonant material in the spheroid (i.e., the halo and, whenever relevant, the bulge) and in the outer disk. Spheroids delay and slow down the initial growth of the bar they host, but, at the later stages of the evolution, they strengthen the bar by absorbing angular momentum. Increased velocity dispersion in the (near-)resonant regions delays bar formation and leads to less strong bars. When bars form they are vertically thin, but soon their inner parts puff up and form what is commonly known as the boxy/peanut bulge. This gives a complex and interesting shape to the bar which explains a number of observations and also argues that the COBE/DIRBE bar and the Long bar in our Galaxy are, respectively, the thin and the thick part of a single bar. The value of the bar pattern speed may be set by optimising the balance between emitters and absorbers, so that a maximum amount of angular momentum is redistributed. As they evolve, bars grow stronger and rotate slower. Bars also redistribute matter within the galaxy, create a disky bulge (pseudo-bulge), increase the disk scale-length and extent and drive substructures such as spirals and rings. They also affect the shape of the inner part of the spheroid, which can evolve from spherical to triaxial.11/2012;
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ABSTRACT: The central, or x1, family of periodic orbits is the most important one in almost all two-dimensional numerical models of galactic bars in the literature. However, we present evidence that in two-dimensional models with sufficiently large bar axial ratios (a/c 6), stable orbits having propeller shapes play the dominant role. In our models this propeller family is in fact a distant relative of the x1 family. There are also intermediate cases in which both families are important. The dominance of one family over the other may have direct consequences on the morphological properties of the bars that can be constructed from them, properties such as face-on bar thinness and strength as well as the boxiness of the outer isophotes.The Astrophysical Journal 12/2008; 624(2):693. · 6.73 Impact Factor
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
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
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.
c ? 2001 RAS
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
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.
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
c ? 2001 RAS, MNRAS 000, 1–11
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.
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 <
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
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
c ? 2001 RAS, MNRAS 000, 1–11
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
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
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
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6P.A. Patsis et al.
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
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
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
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
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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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
c ? 2001 RAS, MNRAS 000, 1–11
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<
=1.5. The third column gives a
short comment about what characterizes the model in order to
facilitate its identification.
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.
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
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
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
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).
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
(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
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
(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).
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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.
This paper has been produced using the Royal Astronomical
Society/Blackwell Science LATEX style file.
c ? 2001 RAS, MNRAS 000, 1–11