Orbital dynamics of three-dimensional bars: II. Investigation of the parameter space

Abstract

We investigate the orbital structure in a class of 3D models of barred galaxies. We consider different values of the pattern speed, of the strength of the bar and of the parameters of the central bulge of the galactic model. The morphology of the stable orbits in the bar region is associated with the degree of folding of the x1-characteristic. This folding is larger for lower values of the pattern speed. The elongation of rectangular-like orbits belonging to x1 and to x1-originated families depends mainly on the pattern speed. The detailed investigation of the trees of bifurcating families in the various models shows that major building blocks of 3D bars can be supplied by families initially introduced as unstable in the system, but becoming stable at another energy interval. In some models without radial and vertical 2:1 resonances we find, except for the x1 and x1-originated families, also families related to the z-axis orbits, which support the bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis of the main bar. This is favoured in the slow rotating bar case. Comment: 11 pages, 20 figures, 1 table, to appear in MNRAS

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arXiv:astro-ph/0204078v1 4 Apr 2002
Mon. Not. R. Astron. Soc. 000, 1–11 (2001)Printed 1 February 2008(MN LATEX style file v1.4)
Orbital dynamics of three-dimensional bars:
II. Investigation of the parameter space
Ch. Skokos,1P.A. Patsis,1E. Athanassoula2
1Research Center of Astronomy, Academy of Athens, Anagnostopoulou 14, GR-10673 Athens, Greece
2Observatoire de Marseille, 2 Place Le Verrier, F-13248 Marseille Cedex 4, France
Accepted .... Received ....; in original form ....
ABSTRACT
We investigate the orbital structure in a class of 3D models of barred galaxies. We
consider different values of the pattern speed, of the strength of the bar and of the pa-
rameters of the central bulge of the galactic model. The morphology of the stable orbits
in the bar region is associated with the degree of folding of the x1-characteristic. This
folding is larger for lower values of the pattern speed. The elongation of rectangular-
like orbits belonging to x1 and to x1-originated families depends mainly on the pattern
speed. The detailed investigation of the trees of bifurcating families in the various mod-
els shows that major building blocks of 3D bars can be supplied by families initially
introduced as unstable in the system, but becoming stable at another energy interval.
In some models without radial and vertical 2:1 resonances we find, except for the x1
and x1-originated families, also families related to the z-axis orbits, which support the
bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis
of the main bar. This is favoured in the slow rotating bar case.
Key words: Galaxies: evolution – kinematics and dynamics – structure
1INTRODUCTION
Barred galaxies have bars of very different strength, ranging
from the weak bars of SAB galaxies to the strong bars of e.g.
NGC 1365 (Lindblad 1999). They may have large, small, or
no bulge components in their centers. The possibility that
bars in late-type barred spiral galaxies end at their inner
Lindblad resonance (hearafter ILR) has also been considered
(Lynden-Bell 1979, Combes & Elmegreen 1993, Polyachenko
& Polyachenko 1994) and this would imply that in some
cases bars may have corotation far beyond their ends. It is
thus important to understand whether and to what extent
the orbital structure changes with the basic parameters in
the models. We investigate this here using a class of models,
the individual representatives of which differ in their central
mass concentration and in the pattern speed and strength
of the bar.
We follow the evolution of all the families of periodic
orbits we think may play a role in the dynamics and mor-
phology of bars and peanuts. We believe we indeed have all
the main families for two reasons. First, the edge-on profiles
of the galaxies are mainly affected by the vertical bifurca-
tions up to the 4:1 vertical resonance. Beyond this resonance
the orbits of the bifurcating 3D families remain close to the
equatorial plane and thus do not characterize the edge-on
morphology. Second, families bifurcated at the n:1 radial
resonances for n > 4 do not in general support the bar (e.g.
Contopoulos & Grosbøl 1989, Athanassoula 1992).
The models presented here are static, but they may be
viewed as corresponding to individual phases of an evolu-
tionary process of the dynamical evolution of a galaxy within
a Hubble time. Therefore, a complete investigation of the dy-
namical system is necessary in order to find all orbits pos-
sibly associated with the presence of specific morphological
features.
In the first paper of this series (Skokos, Patsis &
Athanassoula 2002, hereafter paper I) we presented the basic
families in a model composed of a Miyamoto disc of length
scales A=3 and B=1, a Plummer sphere of scale length 0.4
for a bulge 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 1 Myr and the mass
unit 2 × 1011M⊙.
In the present paper we compare the orbital structure
of our basic model with those encountered in five more mod-
els. Our models, including the fiducial model A1 of paper I,
are described in Table 1. G is the gravitational constant,
MD, MB, MS are the masses of the disk, the bar and the
bulge respectively, ǫs is the scale length of the bulge, Ωb is
the pattern speed of the bar, Ej(r-IILR) and Ej(v-ILR) are
the values of the Jacobian for the inner radial ILR and the
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Ch. Skokos 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, Ej(r-IILR) is the Jacobian for the inner radial ILR,
Ej(v-ILR) is the Jacobian for the vertical 2:1 resonance, 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
Ej(r-IILR)
Ej(v-ILR)
Rc
comments
A1
A2
A3
B
C
D
0.82
0.82
0.82
0.90
0.82
0.72
0.1
0.1
0.1
0.1
0.1
0.2
0.08
0.08
0.08
0.00
0.08
0.08
0.4
0.4
0.4
–
1.0
0.4
0.0540
0.0200
0.0837
0.0540
0.0540
0.0540
-0.441
-0.470
-0.390
–
–
-0.467
-0.360
-0.357
-0.364
–
-0.364
-0.440
6.13
13.24
4.19
6.00
6.12
6.31
fiducial
slow bar
fast bar
no bulge
extended bulge
strong bar
vertical 2:1 resonance respectively, and Rc is the corotation
radius.
This paper is organized as follows: In section 2 we dis-
cuss models with fast, or with slow bars. Section 3 introduces
a model with no 2:1 resonances, section 4 a model with ver-
tical but no radial ILR, and section 5 a model with a massive
bar. We conclude in section 6.
2THE EFFECT OF PATTERN SPEED
2.1A slow rotating bar
Model A2 is the same as model A1 in everything, except for
the pattern speed, Ωb= 0.02, which is less than half that of
model A1. The corotation in this model is at 13.24, rather
than at 6.13 as in model A1, and the outer inner Lindblad
resonance (OILR) is now at 6.1, i.e. roughly the end of the
bar or the corotation distance of the models with Ωb= 0.054.
The changes in the dynamical behaviour are much more
important than a stretching of the corotation radius by a fac-
tor larger than two and an enlargement of the x2-x3 loop in
the characteristic and in the stability curves of the model.
New bifurcations and new gaps are introduced, while the
morphology of some of the existing families changes dras-
tically. The differences are so big as to introduce nomen-
clature issues. Let us start our examination of the main
simple-periodic families and of their bifurcations with the
help of the characteristic diagram for planar orbits, shown
in Fig. 1. Following the convention introduced in paper I, we
draw with a black line the parts of the characteristics which
correspond to stable parts of the families, while grey is asso-
ciated with instability. There are two main characteristics, or
rather families of characteristics. The lower one is confined
to the region below x ≈ 5.5. It is divided from the upper
characteristic by a gap, occurring roughly at Ej = −0.128.
There are are also a number of 3D families bifurcating from
these characteristics, of which the most important ones will
be described at the end of this section.
The main feature of the characteristic diagram is a con-
tinuous curve constituted by the simple-periodic 2D families
x1, x2 and x3. We will follow it counter-clockwise. Starting
close to Ej = −0.5 for x = 0 we walk along the characteristic
of the typical x1 family. The orbits there are elliptical-like
and support the bar.
At the first S →U transition of x1 the family x1v1 is
bifurcated. That means we have reached at this energy the
vertical 2:1 resonance. It has a similar evolution as in the
Figure 1. Characteristic diagram for the 2D families of model
A2. Grey parts of the lines show the unstable parts of the families.
In (b) we give an enlargement of the area included in the frame
in (a).
fiducial case (paper I), but it is complex unstable for a con-
siderably smaller energy range. This affects strongly the ver-
tical profile of the model (Patsis, Skokos & Athanassoula,
2002a hereafter paper III). Since it is a 3D family it is not
included in Fig. 1.
The first radial bifurcation occurs at Ej ≈ −0.31 and
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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space
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gives the family o1. This is stable for a tiny Ej interval, just
after the S→U transition. It then follows a S→U→S→U
sequence and ends again on x1. Thus this family builds a
bubble, both in the characteristic and the stability diagram,
together with x1 or with its indices, as did family t1 in model
A1. Its morphology, however, shows that it is related to a ra-
dial 1:1 resonance (Papayannopoulos & Petrou 1983), since
both cuts with the y = 0 axis are for x > 0 (alternatively
x < 0), so that it can be viewed as a distorted circle. It nev-
ertheless has three tips or ‘corners’, of which the two close
to the y axis are very sharp and for large Ej values they
develop loops. This morphological evolution is reflected by
the small orbits drawn close to the o1 curve on Fig. 1a.
The next S→U transition brings in the system x1v3.
This is a 3D family, so again it is not included in the dia-
gram. When x1 becomes again stable close to Ej ≈ −0.134,
its orbits have developed loops along the major axes of the
ellipses. Since we are already at the area included in the
frame in Fig. 1a, it is easier to follow the evolution of the
families on the characteristic in Fig. 1b. We observe that
close to Ej = −0.115 the curve has a bend and continues
towards lower Ej and higher x values. On the bend x1 orbits
are still very elongated with loops at the y axis, as noted
by a x1 orbit drawn there. The x1 family has developed
these loops well before the bend. Between Ej = −0.115 and
Ej = −0.13, at the rising part of the characteristic, towards
lower Ej values, the x1 orbits become again ellipse-like and
their loops vanish (for the time being we forget about the
gray branch we observe at the same area). Meanwhile, the
characteristic curve has two more bends at x roughly 3 and
5 respectively and for almost the same value of Ej ≈ −0.13,
and then follows the long branch towards lower Ej values,
which reaches Ej ≈ −0.47. On this branch and close to
Ej = −0.13 the x1 orbits have small ellipticities and become
even rounder as we move to lower Ej values. Finally after
Ej ≈ −0.17 the orbits are elongated along the minor axis
of the bar, and are stable (except for −0.23 < Ej < −0.2),
i.e. they belong to the x2 family. At Ej ≈ −0.47 the curve
folds again and continues its journey towards larger Ej val-
ues. The orbits at this branch are typical x3 orbits and exist
until Ej ≈ −0.29, where they change multiplicity. Thus in
this model the x2 and the x3 families are continuations of
the x1, the transition being made by circular and circular-
like orbits, rather than by a gap as in the standard cases
(Contopoulos & Grosbøl 1989, Athanassoula 1992a, paper I
etc.).
At Ej ≈ −0.23 the stability index associated with the
3D bifurcations intersects the −2 axis. So we have the bifur-
cation of a new family with the same multiplicity. We call
this family x2v1. We emphasize the fact that this is a simple
periodic family, since in model A1 (paper I) we had already
encountered a 3D bifurcation of x2 (family x2mul2), which,
however, is of multiplicity 2. Since this new family is a direct
bifurcation of x2 at the S→U transition close to Ej = −0.23,
as we move towards larger values of Ej, it inherits the sta-
bility of the parent family, i.e. it is stable. It stays stable for
a large energy interval, −0.23 < Ej < −0.18, which means
that it is a family that can affect the morphology of the
galaxy. Its morphology can be seen in Fig. 2. As we can see
this family can support a peanut-like feature, which, how-
ever, is elongated not along the major but along the minor
axis of the main bar. If such orbits are populated in a real
Figure 2. Stable orbit of the x2v1 family.
Figure 3. Successive x13 orbits. They are all unstable.
galaxy, then they will support a 3D stellar inner bar with a
‘x2 orientation’.
Close to the part of the x1 characteristic for −0.13 <
Ej < −0.115, where the curve folds and extends towards
lower energies (Fig. 1b), we have, besides the ‘x1 part’, a
gray branch (unstable orbits) that bridges the main loop
with another branch of the characteristic diagram existing
at the same energies and for larger x values. If this bridge
was missing then we would have a classical type 2 gap as
at the radial 4:1 resonance regions (Contopoulos & Grosbøl
1989). What we have now could be called a pseudo-gap. The
orbits of this branch are unstable and belong to a family we
call x13, since it starts for low x values as x1 at point ‘A’
(Fig. 1b) and reaches at ‘B’ a horizontal branch, which is
the characteristic curve of a x3-like family. x13 is a radial
bifurcation in ˙ x, so the curve we give in Fig. 1 for this family
is just the projection of its characteristic in the (Ej,x) plane.
The morphology of these orbits is expected to be related
with inclined ellipses, whose major axis shifts from being
parallel to the bar major axis (for members on or near the
major loop characteristic) to parallel to the bar minor axis
(for members on or near the x3′characteristic). The shift
happens in a small energy interval, in which the x1 orbits
have the longest projections on the y axis. Successive orbits
of x13, as we move from ‘A’ to ‘B’ (Fig. 1b), are given in
Fig. 3. The evolution of the stability indices of x1 in this
area follow every possible complication one could imagine
in order to avoid bifurcating a stable family with similar
morphology. Due to this ‘conspiracy’ it was not possible for
us to find a stable x13-like family.
The second part of the characteristic diagram, at the
same energies as the ‘x1 part’ and for higher x values, has
orbits which are x3-like. These orbits are ellipses elongated
along the minor axis of the bar and are almost everywhere
unstable, except for a tiny part of the characteristic for
Ej ≈ −0.175. We thus called them x3′. Moving along the
x3′branch of the characteristic towards larger Ej values, we
encounter a step-like feature in the curve (Fig. 1b) and be-
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Ch. Skokos et al.
Figure 4. Stable orbits of the families x1′v4 (a), and x1′v5 (b).
yond it we have planar orbits, which can be easily described
as prograde quasi-circular orbits. Their general dynamical
properties and their relation with other families at the area
resemble those of the x1 family. So this family is a kind of
continuation of x1, which we call x1′(as we called, for lower
energies, the continuation of the x3 family x3′). The stability
indices of x1′oscillate and at the tangencies with the b = −2
axis the 3D families x1′v4, x1′v5 etc. are born. We call them
like this because their morphology on the (x,z) and (y,z)
projections resembles the morphology of the x1v4 and x1v5
families of the fiducial case. The bifurcated 3D families re-
main as stable close to the equatorial plane, i.e. they do not
characterize the vertical profile of the model, although they
have large stable parts. It is important to note that in this
case the shape of the x1′orbits – and of the (x,y) projec-
tions of x1′v4 and x1′v5 as well – are not elongated along the
major axis of the bar, but are quasi-circular. Thus, they do
not enhance the bar towards the corotation radius (13 kpc).
This can be seen in Fig. 4.
The characteristic of x1′, as in the case of model A1 for
x1, has a local maximum at Ej ≈ −0.11. At the decreas-
ing branch (lower x’s for larger Ej values) the orbits of the
family develop ‘corners’. The usually rectangular-like orbits
found in the 4:1 region (cf. Fig. 2g in paper I) are for this
model square-like. x1′has a stable part just after the turning
point, while in model A1 the decreasing branch is almost ev-
erywhere unstable. For yet larger energies, when the orbits
at their four apocentra have loops, x1′is unstable.
As can be seen in Fig. 1b the gap at Ej ≈ −0.11 is a
real type 2 gap, the upper branch of which has stable circu-
lar orbits at the ‘increasing x’ part and unstable hexagonal
orbits at the decreasing part following it at larger Ej val-
ues. The latter are not much elongated along the y axis.
Due to this morphological evolution of the x1 family there
are no elliptical-like orbits elongated along the y axis to ex-
tend the bar towards corotation. The elongated orbits which
reach the farthest out in the y direction are elliptical-like or-
bits with loops, reaching y ≈ 9.4, surrounded by a roundish
structure reaching the corotation region (Fig. 5).
Before closing our description of model A2, we should
mention that the family x1v4, initially bifurcated as dou-
ble unstable, becomes stable for larger energies and provides
the system with 3D orbits with low |z|. The orbit we give in
Fig. 6 has Ej = −0.12, while the family x1v4 bifurcates for
Ej ≈ −0.245 at a D→U transition of x1. The evolution of
Figure 5. Stable orbits for model A2.
Figure 6. Stable orbit of the x1v4 family of model A2.
the stability indices of this family in model A2 is less com-
plicated than in model A1. It nevertheless shows all kinds of
instabilities we encounter in 3D Hamiltonian systems and fi-
nally ends again on x1. This means that it can be considered
both as a direct and as an inverse bifurcation of x1.
Summarizing the main differences of the orbital be-
haviour of the slow rotating bar model from that in the
fiducial case, we underline the existence of a complicated
common characteristic of the x1, x2 and x3 families. As a
consequence the simple-periodic families of the x1-tree ap-
pear in two parts. The second part consists of x1′and its 3D
bifurcations. The families of the x1′-tree have large stable
parts, but they do not help the bar reach closer to corotation
since they are quasi-circular (or have quasi-circular projec-
tions on the equatorial plane). The rectangular-like orbits
in this case are almost squares. The model also includes a
simple periodic x2-like 3D family. Other differences in the
orbital behaviour from model A1 that should be mentioned
is the small complex unstable part of x1v1 and the bifurca-
tion of the family x1v4 at a D→U stability transition.
2.2A fast rotating bar
Model A3 has a fast rotating bar. Its pattern speed is 0.0837,
which brings corotation to 4.2 kpc, i.e. closer to the center
than the end of the imposed bar. All other parameters re-
main as in models A1 and A2.
The major effect, as expected, is that the OILR ap-
proaches the IILR, and the size of what we would call ‘x2-
region’ is drastically reduced. In model A3 both x2 and x3
families still exist. The size of the semi-major axis of the
largest x2 orbit is 0.63 kpc. This means, that the x2 orbits
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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space
5
Figure 7. Stability diagram for the family x1 in model A3. The
black bold curves at the right part of the figure (−0.265 < Ej<
−0.235) are the stability indices of the family x1v8. Light grey
curves indicate instability.
could support features of sizes about ≈ 1.2 kpc. In other
words in such models, the x2 family could play a role in the
dynamics of the innermost 1 kpc of the system if the corre-
sponding orbits are populated, despite the fact that the x2-
loops we find are tiny (∆Ej ≈ 0.01) in comparison to those
of models A1 and of course A2. In Fig. 7 we see the evolution
of the stability indices of this model. Note the small ellip-
tical features around Ej ≈ −0.385, which are made from
the combination of the stability indices of x2 and x3. The
stability indices of these two families do not have any other
cuts or tangencies with the b = 2 or b = −2 axes and thus
this model has no 3D families oriented perpendicular to bar
major axis and cannot form a peanut with this orientation.
The oscillations of the b1 and b2 curves of x1 bring
in the system the families x1v1, x1v3 and x1v5 as stable.
Their dynamical behaviour, and thus their importance for
the global dynamics of the system, do not differ from what
we encountered in the fiducial case, and so we do not discuss
it further here. In this model x1v4 is not significant. It re-
mains unstable until its orbits reach high z values above the
equatorial plane. The curves indicated by
the orbits at the branch of the characteristic of x1, after the
bend of the curve towards lower energies for Ej ≈ −0.235
(see Fig. 8 below). Light gray indicates also in Fig. 7 unsta-
ble orbits. The lower index almost goes through the point
of intersection of x1 with the −2 axis. However, because of
the location of the second index, we do not have a loop that
closes on x1 there.
The new elements that the study of this model brings to
the investigation of the orbital dynamics of barred potentials
are focused at the region of the (type 2) gap at the 4:1
resonance. In Fig. 8 we show what is new in this model on a
characteristic (Ej,x) diagram of x1. We have also included
the (Ej,x) projections of a planar family (q0), which has
˙ x0 ?= 0 in the initial conditions, and a 3D family (x1v8),
which is unstable in model A1.
Let us start from the latter. As can be seen in Fig. 7,
the stability index associated with the vertical bifurcations
has its seventh cut with the b = −2 axis at Ej ≈ −0.265
←
x1 correspond to
Figure 8. Part of the characteristic diagram of model A3. It
shows the curve of the family x1 at the 4:1 resonance region, and
the (Ej,x) projections of families q0 and x1v8. Light gray color
indicates unstable orbits.
(the depth and size of the unstable region is very small; we
observe in Fig. 7 that the depth and size of the successive
unstable regions decreases with increasing energy). At this
point a new stable family is born. Fig. 8 shows that this
family is bifurcated just before the local maximum of the
x1 characteristic curve. The (x,z) and (y,z) morphology of
this new family is similar to that of family x1v8 of our fidu-
cial model (cf. Fig. 17c in paper I) and thus, according to
the rules set in paper I, we call it x1v8, although it emerges
at the seventh vertical bifurcation. In model A3 the suc-
cession of appearance of the bifurcating families associated
with the vertical 5:1 resonance is reversed compared with the
families of the corresponding instability strip in the fiducial
case. Now this instability strip is located before the local
maximum of the x1-characteristic at Ej ≈ −0.26 (Fig. 8),
while in model A1 it is located beyond the corresponding
local maximum. As discussed in paper I when the evolution
of the stability index of x1 associated with the vertical bi-
furcations has successive cuts with the b = −2 axis giving
rise to a S→U→S sequence in its stability, a stable and an
unstable family are introduced in the system. In model A1
for all instability strips at the vertical resonances before the
local maximum of the characteristic curve, the families in-
troduced as stable at the S→U transition are bifurcations in
z, and the unstable ones, bifurcated at the U→S part, are
bifurcations in ˙ z. The opposite is true for the 5:1 resonance
instability strip located beyond the local maximum. There
we had a stable family bifurcated in ˙ z, which we called x1v7
and an unstable one bifurcated in z we called x1v8 (paper
I). In the present model the corresponding instability strip
of the vertical 5:1 resonance is located before the local max-
imum of the x1-characteristic for Ej ≈ −0.26 (Fig. 8) and
the family introduced as stable is a bifurcation in z. Since
we keep the nomenclature introduced in the fiducial model
throughout this series of papers, this is family x1v8 and
the bifurcation in ˙ z, unstable in the present model, is x1v7.
We note that while the x1v7 family of model A1 very soon
gets orbits with large |z|’s, family x1v8 is stable everywhere
and its orbits remain confined close to the equatorial plane
(Fig. 9).
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