Issue |
A&A
Volume 495, Number 3, March I 2009
|
|
---|---|---|
Page(s) | 743 - 758 | |
Section | Extragalactic astronomy | |
DOI | https://doi.org/10.1051/0004-6361:200810149 | |
Published online | 22 December 2008 |
Invariant manifolds and the response of spiral arms in barred galaxies
P. Tsoutsis1,2 - C. Kalapotharakos1 - C. Efthymiopoulos1 - G. Contopoulos1
1 - Research Center for Astronomy, Academy of Athens, Soranou
Efessiou 4, 11527 Athens, Greece
2 - Department of Physics, University
of Athens, 11527 Athens, Greece
Received 7 May 2008 / Accepted 4 December 2008
Abstract
The unstable invariant manifolds of the short-period
family of periodic orbits around the unstable Lagrangian points
L1 and L2 of a barred galaxy define loci in the configuration
space, which take the form of a trailing spiral pattern. In previous
works we explored the association of such a pattern to the
observed spiral pattern in N-body models of barred-spiral galaxies
and found it to be quite relevant. Our aims in the present paper
are: a) to investigate this association in the case of the
self-consistent models of Kaufmann & Contopoulos (1996, A&A, 309, 381), which
provide an approximation of real barred-spiral galaxies; b) to
examine the dynamical role played by each of the non-axisymmetric
components of the potential, i.e. the bar and the spiral
perturbation, and their consequences on the form of the invariant
manifolds; and c) to examine the relation of ``response'' models
of barred-spiral galaxies with the theory of the invariant
manifolds. Our method relies on calculating the invariant manifolds
for values of the Jacobi constant close to its value for L1 and
L2. Our main results are the following. a) The invariant
manifolds yield the correct form of the imposed spiral pattern
provided that their calculation is done with the spiral potential
term turned on. We provide a theoretical model explaining the form
of the invariant manifolds that supports the spiral structure. The
azimuthal displacement of the Lagrangian points with respect
to the bar's major axis is a crucial parameter in this modeling.
When this is taken into account, the manifolds necessarily
develop in a spiral-like domain of the configuration space,
delimited from below by the boundary of a banana-like non-permitted
domain, and from above either by rotational KAM tori or by cantori
forming a stickiness zone. On the contrary, if the whole
non-axisymmetric perturbation is artificially ``aligned'' with the bar
(i.e. there is no azimuthal shift of the Lagrangian manifolds), the
manifolds support a ring rather than a spiral structure. b) We
construct ``spiral response'' models on the basis of the theory of the
invariant manifolds and examine the connection of the latter to
the ``response'' models (Patsis 2006, MNRAS, 369, 56) used to fit real barred-spiral
galaxies, explaining how the manifolds are related to a number of
morphological features seen in such models.
Key words: chaos - galaxies: kinematics and dynamics - galaxies: spiral
1 Introduction
The ordered or chaotic nature of orbits in barred galaxies has been the subject of many investigations in the literature (Contopoulos 1981; Pfenniger 1984; Sparke & Sellwood 1987; Pfenniger & Frendli 1991; Kaufmann & Contopoulos 1996; Patsis et al. 1997; Fux 2001; Pichardo et al. 2004; Kaufmann & Patsis 2005). Interest in this problem stems from the fact that the existence (and degree) of chaos has direct consequences on the morphological features of a rotating galaxy. In particular, the appearance of a high degree of chaos in the corotation region is one of the main reasons for the bars terminating near corotation (Contopoulos 1981; see Contopoulos 2002; pp. 473, 474 for a review).
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Figure 1:
a) Projections of the invariant manifolds
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Beyond corotation, prominent structures are commonly observed such as rings or spiral arms. The role of the chaotic orbits in the dynamics of such structures is still a widely open problem, but recently some progress has been made towards its understanding. In particular, a theoretical model has been proposed and numerically explored (Voglis et al. 2006a,b; Romero-Gomez et al. 2006, 2007), according to which the spiral arms (or rings) are supported by the unstable invariant manifolds of the two short-period families of unstable periodic orbits around the unstable Lagrangian equilibria L1 and L2 (called hereafter the PL1 and PL2 families respectively). This theory was extended by Tsoutsis et al. (2008), by examining the contribution of the unstable manifolds of other families, besides PL1 or PL2, to the same phenomenon. The importance of the chaotic orbits in supporting the spiral structure of barred galaxies has also been emphasized by Patsis (2006).
The following is a brief account of the theory of the invariant manifolds:
- 1.
- We consider a 2D approximation of the orbits in the disk plane of
a barred - spiral galaxy, given by the Hamiltonian
In this expression,are polar coordinates in the rotating frame,
,
is the angular momentum in the rest frame, V0 is the axisymmetric potential and V1 is the non-axisymmetric potential perturbation due to the bar and to the spiral arms. The parameter
is the angular speed of the rotating frame, which coincides with the bar-spiral pattern speed in an approximation in which the latter is assumed to be unique.
- 2.
- The Hamiltonian flow under (1) yields two stable
(L4, L5) and two unstable (L1, L2) Lagrangian
equilibrium points (in the rotating frame) at which a star corotates
with the pattern. The unstable manifold
of L1 is defined as the set of all the initial conditions
in the phase space for which the resulting orbit tends asymptotically to L1 in the backward sense of time, namely
wheredenotes the position (point in phase space) at time t of a particle along an orbit starting with the above initial conditions, and the norm
means the Euclidean distance between this point and the phase space point
, corresponding to L1. All the points of the manifold
yield the same value of the Jacobi constant, equal to
. Furthermore, since L1 is simply unstable,
is a two-dimensional manifold embedded in the three-dimensional hypersurface of the phase space corresponding to a fixed Jacobi constant
. Similar definitions and properties hold for L2 and
and for the stable manifolds
,
, i.e. the sets of initial conditions tending asymptotically to L1, or L2 in the forward sense of time, as
.
- 3.
- For
, a short-period unstable periodic orbit (PL1) bifurcates from L1 (and the symmetric orbit PL2 from L2). This orbit forms a small loop around L1 (Fig. 1a, thick solid curve), which corresponds to a 1D-torus in the phase space. This torus is ``whiskered'', i.e., it possesses its own asymptotic manifolds. In particular, the unstable manifold of PL1 is now defined as
where the notationrefers to the minimum of the distances of Q(t) from the locus of all the phase space points of the orbit PL1. For any fixed value of
,
is a two-dimensional manifold embedded in the three-dimensional hypersurface of constant
. Figure 1a shows the projection of a small part of this manifold, close to PL1, in the configuration space
,
. This is drawn approximately, by calculating a number of orbits with initial conditions on
, and close to PL1. The possibility to find such initial conditions is guaranteed by the fact that the manifold
is tangent to the unstable manifold of the linearized Hamiltonian flow near PL1 (the so-called Grobman 1959; and Hartman 1960 theorem), and the latter is calculated by diagonalizing the Floquet matrix of the orbit PL1. In Fig. 1a we draw the part of the manifold lying outside corotation for a particular model of barred galaxy. We can see that close to PL1 the orbits form epicyclic loops of size nearly equal to the PL1 loop, while, in the same time, the guiding center recedes from PL1 along a path which yields a trailing spiral arm. An analysis of the linearized flow yields that the deviation of the guiding center from PL1 is exponential in time, with a rate determined by the positive characteristic exponents of the Floquet matrix of PL1. Furthermore, in generic galactic potentials all the orbits on
are chaotic. (The same phenomena hold for the orbit PL2 and the manifold
also plotted in Fig. 1a).
- 4.
- In strongly nonlinear models (as is the case of strongly barred
galaxies with conspicuous spiral arms), further away from PL1 the
size of the epicycles becomes great (it may exceed the size of the
bar). Such an example is shown in Fig. 1b, referring to the orbits of
the
family in a N-Body model of a barred galaxy (Voglis et al. 2006a,b). We see that one such orbit (bold) forms two relatively small loops near PL1, reaching the apocentric positions A1 and A2, but the exponential recession of the guiding center is so fast that there is no loop formed between the second and third (A3) apocentric positions. Furthermore, the fourth apocentric position is at a distance about twice the bar's major semi-axis. Further integration beyond that of Fig. 1b shows that, in fact, all these orbits belong to the so-called ``hot population'' (Sparke & Sellwood 1987), i.e., the orbits make several consecutive oscillations in and out of corotation. Kaufmann & Contopoulos (1996, their Fig. 21a) suggested that such orbits can partly support the bar and partly the spiral arms.















Returning to the role of the invariant manifolds, the intersection
of
with the apocentric surface of section yields
an one-dimensional locus of points. Such a locus can be projected on
either the phase portrait plane
or the
configuration space
.
Figure 1c shows the latter
projection in the case of the same manifold as in Fig. 1b, but
calculated for a much larger length. Every point in Fig. 1c
corresponds to one apocentric position of a chaotic orbit with
initial conditions on the unstable manifold. Clearly, the apocentric
positions along
yield a locus which also supports
a trailing spiral arm over, however, a much larger extent of
than in the case of Fig. 1b. The manifold of Fig. 1c takes
a typical form known in dynamical systems' theory to be associated
with the so-called phenomenon of homoclinic chaos. Briefly,
the manifold develops lobes forming oscillations close to the
apocentric points of the periodic orbits PL1 or PL2. Such
oscillations are analyzed in detail in the sequel.
We should stress that an analysis of the Floquet matrix of the PL1
or PL2 families yields that only the directions of the unstable
invariant manifolds
,
are such
as to define trailing spiral arms, while, close to L1 or L2,
the stable manifolds
,
define
leading spiral arms. Furthermore, in the forward sense of time the
chaotic orbits are attracted in directions of the phase space along
the unstable manifolds. In the sequel we no longer refer to the
stable manifolds
,
,
and the
term ``invariant manifolds'' always implies the unstable manifolds
,
.
In summary, the theory of the invariant manifolds, viewed as either
the loci on which lies the continuous flow of a swarm of orbits
(Romero-Gomez et al. 2006, 2007), or the loci of apocentric
positions of these orbits (Voglis et al. 2006a; Tsoutsis et al.
2008), predicts the formation by the manifolds of a trailing spiral
pattern beyond corotation. Naturally, the central question that
should be posed now is whether (and up to what extent) the spiral
arms formed self-consistently in real galaxies can be associated
with the spiral patterns formed by the invariant manifolds
.
In our previous works (Voglis et al. 2006a; Tsoutsis
et al. 2008), we examined this question by considering the spiral
arms formed in an N-Body model of a barred galaxy and found such
an association to be quite relevant.
In the present paper, our main goal is to examine the same question
in simple models of real barred-spiral galaxies for which some
reliable estimation of both the gravitational potential and the
pattern speed have been provided in the literature by methods
independent of the previous considerations. To this end, we selected
the potential models and pattern speeds reported in the study of
Kaufmann & Contopoulos (1996) for three real galaxies, NGC 3992,
NGC 1073 and NGC 1398. This choice is motivated by the fact that
Kaufmann & Contopoulos (1996) constructed approximate
self-consistent models of the studied galaxies based on the response
density of the superposition of many stellar dynamical orbits. Thus,
their study yielded not only plausible values of the potential
parameters, or the pattern speed, but also the decomposition of the
potential into components, i.e.,
,
,
and
.
This allows us to check the role of each of these
components, in particular of the non-axisymmetric ones
and
,
in the theory. It should be noted that the
self-consistent technique, pioneered by Schwarzschild (1979), has
been used extensively to provide reliable models of galaxies,
despite the fact that there is no a priori guarantee of the
stability of such models that should ideally be probed via N-body
simulations (see e.g. Smith & Miller 1982).
Besides re-confirming that the invariant manifolds do correlate well with the spiral arms found in the self-consistent models of Kaufmann & Contopoulos (1996), our investigation led to a second non-trivial result analyzed in detail in the sequel: in all three models the bar component is dominant over the spiral component within a large radial extent, but not in a narrow zone beyond corotation. This implies that if one uses only the bar component to calculate the manifolds, the latter yield ring rather than spiral structures. Furthermore, if one adds the spiral perturbation to the potential, but gives no azimuthal tilting to the associated m=2 Fourier component, the manifolds become more open as regards their radial extent, but remain quite symmetric as regards their orientation with respect to the bar's major axis, thus still defining rings rather than spiral arms. Only when the azimuthal deformation of the equipotential surfaces due to a really spiral-like perturbation is taken into account (in the Kaufmann & Contopoulos 1996 paper this was modeled as a simple logarithmic spiral), the manifolds are found to follow closely the spiral arms of the self-consistent models. In some numerical experiments (see Sect. 3 below) we managed to obtain a kind of spiral pattern formed by the initial segments of the invariant manifolds in pure bar models, having, however, to drastically depart from the bar parameters given in Kaufmann and Contopoulos' self-consistent models, and pushing the bar's amplitude to highly non-physical values. But even in that case, the manifold-induced spiral arms are quite different from the spiral arms of the self-consistent models, and they disappear when the manifolds are computed for a longer length. Such an investigation demonstrates that while in principle the strength of the quadrupole moment of the bar's potential causes a ``thickening'' of ring structures, thus facilitating the phenomenon of appearance of spiral arms (Romero-Gomez et al. 2007), this parameter is not sufficient in order to characterize this phenomenon. The azimuthal displacement of the Lagrangian points is the most important parameter. This result probably provides a dynamical basis for understanding the reported failure of pure bar models to reproduce the inner spiral arms emanating at the ends of bars in both particle and hydrodynamical simulations of barred galaxies (e.g. Lindblad et al. 1996; Aguerri et al. 2001).
The paper is organized as follows: Sect. 2 gives the form of the invariant manifolds in the models of Kaufmann and Contopoulos. We examine the manifolds a) when the spiral perturbation is turned-on, and b) in an ``aligned model'' version in which the whole non-axisymmetric perturbation is artificially aligned to the bar. In case (a) the manifolds yield a spiral response, while in case (b) they yield a ring-like response. Since in all the above models the spiral perturbation is strong, we also examine two models corresponding to a ``mean'' and ``weak'' spiral amplitude, created by suitably varying the parameters of some of the original models of Kaufmann & Contopoulos (1996). We finally provide a theoretical justification of the importance of the azimuthal displacement of the Lagrangian points in the form of the invariant manifolds. Section 3 discusses the connection between the theory of the invariant manifolds and the ``spiral response'' models constructed via iterative methods. In particular, we propose a method of constructing response models on the basis of populating by matter the manifolds generated by a ``pure bar'' model. We also calculate response models via the method proposed by Patsis (2006) and discuss a number of morphological features of these models which find a straightforward explanation by the invariant manifolds. Section 4 summarizes our conclusions.
2 Model and invariant manifolds
2.1 Model
The model of Kaufmann & Contopoulos (1996) consists of a number of potential/density terms representing various components of a barred-spiral galaxy. In particular we have:- -
- a halo density term given by a Plummer sphere
- -
- A disk surface density given by an exponential law
- -
- A Ferrers bar with major axis aligned with the y-axis
with
- -
- and a spiral perturbation in the potential
where
and

The latter expression introduces a smoothing of the potential yielding a difference with respect to Eq. (8) which is 10% at the distance


Table 1:
Parameters of models A, B, C (from Kaufmann & Contopoulos
1996). The units are
for
A,
for
,
,
and
,
for r1, r2,
,
a, b, c, and
,
for
,
,
for
,
and
/pc2 for
.
Model A' has the same
parameters as model A, except for the spiral amplitude A=1000 and
the pitch angle
.
Model B' has the same parameters
as model B except for A=2500,
,
,
.
A model is specified by a set of values for the parameters ,
,
,
,
,
a, b, c,
,
i, A, Ar,
,
r1,
,
and r2, as well
as the value of the pattern angular speed
.
In Kaufmann &
Contopoulos (1996), the parameters were adjusted so as to
produce three different self-consistent models which present some
features of three real barred galaxies. The criterion for
self-consistency was that the ``response density'', i.e., the density
obtained by the superposition of many orbits in the fixed potential
should match as closely as possible the imposed density represented
by the above equations. The matching refers to a) the amplitudes of
the surface density map on the disk plane, and b) the phases of the
maxima of the bar and of the spiral arms, in the imposed and in the
response models.
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Figure 2: The ratio of non-axisymmetric forces, due to the bar or to the spiral arms, versus total axisymmetric force, as a function of the distance R along the x-axis (solid) or y-axis (dashed) for models A, B, and C (panels a), b), and c) respectively). The vertical dashed lines mark the distance of the L1 or L2points in each case. |
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The parameters for the three models are given in Table 1. In
the sequel we refer to these as model A, B, and C. The value of
Ar for all three models, as well as the values of r2 and
for model C are missing from Kaufmann & Contopoulos
(1996), where, however, it is noted that any (small) value of Ar,
or of r1, r2 and
does not influence the
self-consistency (provided that r2 is at the end of the spiral
arms). For consistency with the remaining models, we have set
and r2=10 kpc in the case of model C, and
Ar=0 in all three models.
Models A, B, C present some features of the galaxies NGC 3992, NGC 1073, and NGC 1398 respectively. As discussed below, the bar-spiral strengths induced by the parameters of model C are quite untypical of barred-spiral galaxies, although still in the range allowed by observations. Thus, while the theory of the invariant manifolds worked well in all three models, we discuss in detail models A and B, and only some exceptional features of model C interesting for dynamics. It should be pointed out that, while the choice of model parameters was partly based on observations (see Sects. 2, 3 of Kaufmann & Contopoulos 1996), the so-obtained models are only rough representations of the referenced galaxies. For example, images of the galaxy NGC 3992 (e.g. in the I band, Tully et al. 1996) indicate the presence of at least one more arm of amplitude comparable to the main bi-symmetric pattern. Images of the galaxy NGC 1398, (e.g. in the R-band, Hammed & Devereux 1999) reveal the existence of an inner ring structure which has no clear-cut separation from the main spiral structure. Such morphological features are not captured by the potential/density model given by Eqs. (4)-(10). Finally, the use of a n=2Ferrers bar model implies a steep drop of the bar force beyond the bar's limit which would be smoother in a n=0 or n=1 model, and it does also not account for a rectangular-like outline that is observed in many real bars.
These facts notwithstanding, the choice of potential parameters and pattern speeds as in Table 1 ensures the existence of a self-consistent solution for the response density, a fact which would by no means be implied in an arbitrary choice of potential model. Although we do not make explicit use of the library of orbits of the final solution in the present paper, and also no guarantee for the stability of the models is provided in Kaufmann & Contopoulos (1996), the self-consistency property suggests that the spiral arms found in these galaxy models can be stellar dynamically supported. This conclusion is independent of the theory of the invariant manifolds, thus the latter theory can be tested against this conclusion.
The relative importance of the various non-axisymmetric components
of the force with respect to the axisymmetric force vary with the
distance from the center, as can be inferred from Fig. 2. The bar
contributes to the forcing by both an axisymmetric and a
non-axisymmetric component. The axisymmetric component is found as
the azimuthally averaged radial bar force
where










In model A (Fig. 2a) the bar yields the dominant
non-axisymmetric perturbation at all distances up to a zone around
corotation (shown as a vertical dashed line at
). The maximum amplitude of the
non-axisymmetric bar force is 0.32, corresponding to a peak of the
curve at
kpc (in all the panels of Fig. 2 the
innermost local maxima or minima of the curves
at
kpc are artificial, due to the weakening of the
axisymmetric forces which, for a Plummer sphere, are exactly equal
to zero at R=0). The inner width of the zone is found by the point
where
,
which is at a distance
kpc. Beyond that distance, the spiral term dominates over the
bar term, reaching a maximum amplitude equal to 0.21 with respect
to the axisymmetric background. The oscillations of the spiral force
beyond corotation are due to the logarithmic dependence of the
argument
in (7) on r, a fact causing successive
maxima and minima of the spiral force at successive periods of
length
of the argument
.
The first maximum, around
corotation, is the most important. The width of the oscillation from
this maximum to the next defines an approximate value of the radial
wavelength of the spiral density wave, which is
kpc.
In model B (Fig. 2b) the maximum amplitude of the
non-axisymmetric bar force reaches the value 0.75 (for
at
kpc), implying that the bar is quite strong inside
corotation. Nevertheless, even in this galaxy the spiral force
becomes dominant over the bar's non-axisymmetric perturbation around
and beyond corotation. The zone around the first maximum of the
spiral force defines a radial wavelength of the spiral density wave
kpc. The first peak of the spiral force is again
found to be at a distance very close to the corotation radius
r=rL1=3.43 and the amplitude of this peak is 0.35.
Finally, in model C (Fig. 2c) the spiral perturbation near and
beyond corotation reaches such a high amplitude (maximum = 0.57),
that it becomes even stronger than the maximum amplitude of the
bar's perturbation (0.3 for
at
kpc) which takes place well inside corotation. Furthermore, the
spiral arms are tightly wound (the radial wavelength is estimated as
kpc), and the spiral arms extend to cover about
one azimuthal period
.
Thus, model C is exceptional and
will not be discussed in detail in the sequel. Only a feature of
this model interesting for dynamics is discussed in Sect. 2.3.
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Figure 3:
The |
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The bar-spiral amplitudes of models A and B define strongly
nonlinear models, which are above the average but inside the range
of bar-spiral strengths found by recent observations (Laurikainen &
Salo 2002; Buta et al. 2005). In the latter works the maximum value
of the ratio of the tangential force versus the radial force for the
bar and spiral components is denoted by
and
respectively. The average values for SB galaxies in the sample of
Buta et al. (2005) are
and
.
Model A yields
and
.
Model B yields
and
,
both values being a factor 1.3 larger than the specific estimates
reported by Buta et al. (2005) for the galaxy NGC 1073 (
,
), to which model B is associated. In order to have a more
representative sample of models in which the theory of the invariant
manifolds is to be tested, two ``weak'' models, A' and B', are
also considered, which were created by varying some parameters of
models A and B. In model A' the spiral amplitude is A=1000, i.e.
half the value of model A (see Table 1). In model B' the bar mass
is
and the spiral amplitude A=2500. Since these changes
are rather arbitrary, there is no guarantee of self-consistency of
the new models. However, a rough criterion of self-consistency
(Sect. 2.2) can be established if the pitch angle is also
slightly varied in both models (i0=-90 in model A' and
i0=-80 in model B'). The resulting
and
values are
,
for model A' and
,
for
model B'. The spiral strengths of models A', B' are well below
the average of the Buta et al. sample for SB galaxies. Thus, the
mean values
and
of the four models A, B, A', B'become both representative of the average values found in the
observations. On the other hand, the value
of model C is
untypical although still in the range of the observations (Fig. 3).
2.2 Phase portraits and invariant manifolds
The first result of the analysis of the invariant manifolds
can now be demonstrated with the help of Figs. 4 to 6. Figure 4a
shows the phase portrait (surface of section
corresponding to the apocentric positions
,
)
in the case of model A, for a value of the Jacobi
constant
,
which is close to the value
.
Figure 4c shows the same portrait
for
in a so called ``aligned spiral'' version
of model A in which the angle
in Eq. (10) is
replaced by
throughout the whole radial extent of the
spiral arms. This means to artificially ``align'' the spiral arms as
extensions of the bar along the latter's major axis. By this
way we measure the effect of only increasing the amplitude of the
non-axisymmetric perturbation on the form of the invariant
manifolds, while in the original model the manifolds are affected
both by the strength of the non-axisymmetric perturbation and by the
azimuthal displacement of the unstable Lagrangian points with
respect to the bar's major axis.
In Figs. 4a,c the points marked PL1, PL2 correspond to the fixed
points of the PL1 or PL2 short-period orbits which are close to the
positions of the unstable equilibria L1, L2. Furthermore, the
thick dots show the intersection of the unstable manifolds
and
with the surface of section. In
order to facilitate the reading of these diagrams, we note that, for
pr=0 (apsides), beyond some radius
kpc,
Eq. (1) yields that r increases nearly monotonically
with
in all azimuthal directions of a fixed angle
(a small reversal of this monotonic relation, due to the
non-axisymmetric potential terms, is only observed at angles
and in a small interval of radii, of width
kpc around r=5 kpc; the monotonic relation is
re-established after crossing this interval). Thus, in Figs. 4a,c the
semi-plane of the phase portrait with
means
apocentric positions outside corotation, while
means apocentric positions inside
corotation. Note that in this and in all subsequent plots of phase
portraits the values of
are normalized with respect to
the value
,
corresponding to the angular momentum in
the rest frame of a circular orbit at a radius r=a under the action of only the axisymmetric potential.
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Figure 4:
a) Phase portrait near corotation (
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Figure 5:
Same as in Fig. 4, but for model B. The Jacobi constant
is
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Figure 6:
a), b) Same as in Figs. 4a,b but for the model A', and
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The main remarks about the comparison of the two phase portraits are now the following:
- -
- in both portraits chaos is pronounced inside corotation (for
), and the domain of inner invariant KAM curves is deeply inside the bar (at values of
about or below 0.25). Such extended chaotic domains are responsible for the termination of the bar;
- -
- outside corotation (for
), a layer of outer KAM curves has been destroyed in both portraits. This is caused mainly by the growth of the chaotic layer around the unstable -6/1 periodic orbit, which produces a resonance overlap with the chaotic layer of the PL1,2 unstable periodic orbit (negative signs indicate a resonance outside corotation, for which the motion is retrograde in the azimuthal direction). As a result, the chaotic domain extends up to values of
, and the first rotational KAM curves appear a little inside the -4/1 resonance. The domain around the outer Lindblad resonance is almost entirely filled either by rotational KAM curves or by ``resonant'' curves around the -2:1 stable periodic orbits. The islands of stability of the -2:1 resonance have a larger width in the ``aligned spiral'' model (Fig. 4c) because by aligning the spiral perturbation the amplitude of the total non-axisymmetric perturbation increases effectively at large distances from corotation (the width of resonances scales as a power-law of the non-axisymmetric perturbation (see e.g. Contopoulos 2002);
- -
- the white circular domains devoid of points, embedded in the
chaotic sea of both portraits, correspond to prohibited domains of
motion, for
and for the selected values of the Jacobi constant. Such domains exist when
(equal to
);
- -
- the most important difference between the two portraits is
that in the case of the true spiral term turned on (Fig. 4a) the
prohibited domains lose their azimuthal symmetry with respect to the
values
(equal to
), or
referring to the positions of the stable Lagrangian points L4 and L5. Such a symmetry is perfect in the aligned spiral case (Fig. 4c). The limiting boundaries of the prohibited domains are denoted by ``LC4'', ``LC5'' in Fig. 4. The azimuthal deformation of the prohibited domains corresponds to an azimuthal deformation of the associated banana-like prohibited domains appearing in the configuration space (i.e. the disk plane). These domains are similar but should not be confused with the domains delimited by the zero velocity curves of the effective potential in the rotating frame, i.e.
.





On the other hand, the manifolds of the ``aligned spiral'' version of model A (Fig. 4d), calculated up to a length comparable to that of the manifolds of Fig. 4b, show no support of a spiral structure, but only yield a thick ring-like structure. The thickness of the manifolds of Figs. 4b,d is determined by the degree of chaos in Figs. 4a,c. The degree of chaos is determined by the amplitude of the non-axisymmetric perturbation. This is expected from dynamical systems theory, since the overlapping of resonances, which is the main source of production of chaos, depends on the width of the different resonant layers near corotation, which, in turn, depends on only the amplitude of the perturbation.
A more elaborate analysis (Sect. 2.3) shows that, while the outermost radial limit of the invariant manifolds is posed by the existence of absolute barriers, i.e. rotational KAM tori (marked ``KAM'' in Fig. 4b), more stringent limits are practically posed by partial barriers, i.e. cantori, which limit the diffusion within a chaotic zone. Provided these limits, the azimuthal deformation of the invariant manifolds is the crucial factor for the production by them of response spiral arms. This, in turn, is determined by the form of the limiting boundaries LC4 and LC5. The theoretical derivation of these boundaries is given in (Sect. 2.3).
Figure 5 shows the same phenomena in the case of model B. The
qualitative resemblance between Figs. 4a,b and 5a,b is obvious,
although the azimuthal deformation of the limiting boundaries
LC4 and LC5 is more pronounced in Fig. 5b than in Fig. 4b. Also
in this model the manifold exhibits a bridge starting at an angle
clockwise from L1 or L2 (points A,
A'), as well as inner spurs connecting segments of it along both
spiral arms and along the border of the bar. Another feature of
Fig. 5b is that the inner branch of the invariant manifold (inside
the bar) is developed in a domain occupying about one fourth of the
total extent of the bar. This implies that a substantial part of the
bar in the domain near corotation is supported by chaotic orbits. In
fact the non-axisymmetric forcing in model B is much stronger inside
corotation than in model A, a fact causing the destruction of
all the inner KAM curves down to
(Fig. 5a). Such a
type of chaos may lead to a number of observational consequences,
photometric and kinematic, a list of which have been enumerated by
Grosbøl (2003). Finally, the azimuthal deformation of the maxima
of the spiral term with respect to the bar's major axis also turn
out to be the crucial factor for the production by the manifolds of
response spiral arms. In fact, by comparing Figs. 5a,b with the
respective figures in the ``aligned spiral'' version of model B (Figs. 5c,d) we see that the manifolds in the latter case
present some asymmetry as well as a large thickness, due to the high
value of the non-axisymmetric perturbation, but they still largely
deviate from the spiral pattern (gray locus), which was closely
followed by the manifolds of the non-aligned model (Fig. 5b).
Figure 6 shows the phase portrait structure near corotation in
the ``weak spiral'' models A' (Fig. 6a) and B' (Fig. 6b). The
invariant manifolds
and
are
also plotted, and the counterparts of these plots in the
configuration space are shown in Figs. 6b and 6d respectively. As
expected, in both models chaos is considerably reduced with respect
to the strongly nonlinear models A, B, and it is only limited in a
narrow zone in the corotation region. Further away, the phase space
is filled by invariant tori which occupy most of the phase space
volume already at the -4:1 resonance.
The
value of model A' is
,
and this is its only
difference with respect to model A, which has
.
The
thickness of the invariant manifolds is thus reduced with respect to
the thickness of the manifolds of model A (compare Figs. 6b and 4b).
However, the azimuthal deformation of the manifolds is still large
enough to fit the locus of the imposed spiral arms up to an angle
clockwise from L1 or L2, i.e. the manifolds
support quarter turn spiral arms. In the case of model B' (Fig. 6d)
we have
,
which is close but still below the average value
of the Buta et al. (2005) sample. At the value
the region of homoclinic chaos formed by the lobes of the
manifolds near L1 or L2 is already well developed, and the
``inner spurs'' are clearly distinguishable. In fact, in model B' a
small adjustment of the pattern speed (
instead of
32.5 km s-1 kpc-1, as was in model B) yielded the best fit of the
invariant manifolds to the imposed spiral arms. Such a fit can now
be considered as a rough criterion of self-consistency. We see that
the formation of bridges in the manifolds of Fig. 6d result in that
the higher order lobes of the manifold make oscillations which
enhance the density along the manifolds' unstable directions all the
way from L1 or L2. Thus the manifolds support, again, the
imposed spiral arms in a self-consistent way. In fact, since the
overall thickness of the domains covered by the invariant manifolds
increases, the manifolds support the spiral structure up to an angle
larger than
,
i.e. along a length larger than in
model A'.
In conclusion, there is a clear morphological continuity of the
structures produced by the invariant manifolds, from rings to
quarter turn spirals, and then to fully developed spiral arms, as
the value of
increases. The azimuthal displacement of the
unstable Lagrangian points is responsible for the manifolds
producing a spiral-like response, while the increase of the
amplitude
extends the support of the spiral structure to
higher angles
.
Such a morphological continuity of the
invariant manifolds is suggestive of it being a real morphological
feature of barred galaxies.
2.3 Theoretical modeling
![]() |
Figure 7:
The curves of zero velocity (equipotential curves of
the effective potential
|
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A first estimate of the azimuthal deformation of the limiting
boundaries LC4 and LC5 can be done by calculating the azimuthal
displacement of the unstable Lagrangian points L1,2 when
is turned on (Figs. 7a,b, schematic). This can be judged
from the form of the equipotential curves (called hereafter the
``curves of zero velocity'', CZV) in the rotating frame, i.e., the
level curves of
For a logarithmic spiral, the azimuthal shift of L1 is given by
where



The equipotential curves of
do not provide the strictest
limit of allowed motions on the apocentric surface of section
,
or
,
for pr=0,
.
The form of the limiting curves in the configuration space is
shown schematically in Fig. 7c (curves LC4 and LC5, encircling the
stable Lagrangian points L4, L5). These curves are derived by
the requirement that, for any fixed angle
and angular momentum
,
the curve of the function
be tangent to the line



The limiting values of




The limiting curves LC4 and LC5 in the configuration space, derived from Eqs. (14) and (15) are outside the limiting curves provided by the CZVs defined through Eq. (12). This is due to the fact that these limits now refer to

The gray domain in Fig. 7c shows the permissible apocentric
positions of the orbits under a fixed value of the Jacobi constant
close to the corotation value. The interior gray domain between LC4
and LC5 roughly marks the extent of the bar. On the other hand, the
positions of the PL1 and PL2 points are in very narrow strips of
permissible apocentric positions separating the inner right part of
the LC4 curve from the outer right part of the LC5 curve and vice
versa. The invariant manifolds
emanating from these
points necessarily follow the narrow strips leading to the outer
gray domain, thus they yield locally the form of spiral arms.
![]() |
Figure 8:
The resonant phase space structure in the corotation
region in the cases of a) model A with
|
Open with DEXTER |
The evolution of the invariant manifolds further away from PL1 or
PL2 is determined by the resonant structure in the outer corotation
zone. The existence of many resonances accumulating in a narrow
range of distances near the corotation radius causes a chaotic layer
in this region, formed by the mechanism of resonance overlap.
Figure 8 makes a zoom to the phase portraits of the models
considered, in order to demonstrate the relevant phenomena.
Figures 8a,b are zooms to the phase portraits of Figs. 4a, and 5a,
referring to the models A and B. In both cases we find a chaotic
layer extending up to
,
which is delimited by a
rotational KAM torus at
(marked KAM). This torus is
just below the -4/1 resonance, which is stable at the value of the
Jacobi constant
.
On the other hand, most
islands of stability of resonances -m/1, with m>4, which are
closer to corotation, have been destroyed. Their destruction is
followed by the destruction of KAM tori with irrational rotation
numbers, which, according to the standard theory, are transformed
into cantori. Such cantori limit the chaotic flux through their
gaps, and this fact causes some stickiness in a zone very close to
the PL1 and PL2 fixed points. Stickiness phenomena of this type have
been explicitly demonstrated and studied in simple models of the
dynamical systems theory (see e.g. Efthymiopoulos et al. 1997;
Contopoulos et al. 1999; Contopoulos & Harsoula 2008), and they
have also been observed in our N-Body simulations of barred-spiral
galaxies (Tsoutsis et al. 2008). The main outcome of these studies
is that the cantori in a large chaotic sea act as partial barriers
slowing down considerably the rate of escape (or the diffusion) of the chaotic
orbits with initial conditions along or near an invariant manifold.
In the case of the manifolds
plotted in
Figs. 8a,b, which are calculated from 21 iterations of an initial
segment of length
ds=10-4 close to PL1 or PL2, we see that the
manifolds fill only partially the chaotic domain up to the torus
marked KAM. The inner dark region covered by the first iterations of
the invariant manifolds defines a domain called ``inner stickiness
zone'', the projection of which in the configuration space is shown
schematically as a dark gray domain in Fig. 7c. The remaining part of
the chaotic domain up to the curve marked ``KAM'' corresponds
essentially to the light gray domain of Fig. 7c. This domain is
eventually covered by the invariant manifolds after a very large
number of iterations. For example, the manifolds of Figs. 8a,b have
not yet reached the curve KAM after about 50 iterations, which in
both models correspond to about 60 pattern rotation periods. Thus,
during all this time interval the manifolds support a spiral
structure.
The stickiness phenomena keep playing a significant role even when
the spiral perturbation is pushed to untypically high values. For
example, in model C, (Figs. 8c,d) the spiral strength is
,
and under such a high value all the rotational KAM
curves are destroyed, at least up to the outer Lindblad resonance.
Then, while in principle there is no absolute barrier to chaotic
diffusion up to very large distances from corotation, a plot of the
invariant manifolds (Fig. 8c) shows that these manifolds
exhibit again stickiness phenomena, and they practically remain
confined for very large times below the -4/1 resonance (which is
still stable at the value of the Jacobi constant
,
yielding four tiny islands embedded in the large chaotic sea
of Fig. 8c near the level
.
This results in that
the manifolds in the configuration space (Fig. 8d) yield the
form of tightly wound spiral arms. In this particular example, the
domain covered by the invariant manifolds practically coincides
with the ``inner stickiness domain'' of Fig. 7c.
3 Spiral arms as the response of invariant manifolds to bars
![]() |
Figure 9:
A ``spiral response'' model based an initially ``pure
bar'' version of model A. a) Manifolds of the pure bar case for
|
Open with DEXTER |
One immediate consequence of the analysis of the previous sections
is that one cannot induce the morphology of the spiral arms,
corresponding to a particular morphological type of bar, by
calculating the invariant manifolds of the PL1 and PL2 families in
only a pure bar potential. In fact
is most important
near corotation and it must be taken into account self-consistently
in all studies related to the morphology of the spiral arms via the
calculation of invariant manifolds. This result is in agreement and
probably provides a dynamical basis for understanding the results of
both particle and hydrodynamical simulations (Lindblad et al. 1996;
Aguerri et al. 2001) which have reported the inefficiency of
simulations of pure bars to reproduce a spiral structure.
On the other hand, the theory of the invariant manifolds suggests that the spiral arms are linked dynamically to the bar. A plausible scenario for establishing such a link is one in which the bar initiates the process of a spiral response, which is then enhanced self-consistently by the growing contribution of the spiral potential.
![]() |
Figure 10:
Unstable invariant manifolds of the PL1 and PL2 orbits in a
pure bar version of model A in which the bar's mass and pattern
speed are altered with respect to the reference values
|
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![]() |
Figure 11:
Model A for the value of the Jacobi constant
|
Open with DEXTER |
- i)
- we first calculate the invariant manifolds produced by the pure
bar model;
- ii)
- we assume that the invariant manifolds produced by the pure bar
alone ``trigger'' the formation of a ring-like or spiral pattern by
attracting matter along the invariant manifold. The density
gradient along the manifold cannot be uniform, since (a) the speed
of chaotic diffusion is smaller close to L1 or L2 than far
from these points, and (b) the higher order lobes of the invariant
manifolds return to the neighborhood of L1 and L2 (Sect. 2.2).
In order to model the mass distribution along the invariant
manifolds, we consider a number
of small Plummer spheres, of mass
and softening radius
, placed along the invariant manifold of the pure bar case (Fig. 9a) with a linearly decreasing mass from L1 or L2 counterclockwise, namely the mass of the ith particle is given by:
withfrom L1 to L2 counterclockwise and similarly along the symmetric manifold from L2 to L1.
is an estimate of the total mass on the spiral arms (see below). The choice of a linear mass decrease as in (16) along the response spirals is rather arbitrary and it does not follow directly from the theory of the invariant manifolds. However, it does capture the essential feature that the density of points should in general decrease along the unstable manifold as we recede from the unstable periodic orbit.
In the simulation of Fig. 9, we start from the invariant manifolds of the pure bar version of model A, and set
,
, and the mass of each particle fixed so that the total mass of all the particles is equal to
given by
where the quantityis an approximate expression for the density perturbation corresponding to the potential (7) given by the WKB ansatz:
where, according to the formula given by Vandervoort (see Contopoulos & Grosbøl 1988; the used values of
are given in Table 1). Thus,
represents an estimate of the total mass contained in the spiral arms of model A. A gray scale plot of the total surface density corresponding to the above mass distribution is shown in Fig. 9b;
- iii)
- the gravitational potential is calculated anew taking into
account the ``response'' potential produced by the Plummer masses
positioned as described in step (ii). This also yields a new
position of the Lagrangian points L1, L2 as well as a new form
of their invariant manifolds. The procedure should be repeated until
convergence of the positions of the Lagrangian points and of their
manifolds towards a final form is obtained. In practice, we
find that just one iteration suffices to obtain invariant manifolds
which fit the imposed spiral perturbation up to an azimuth
(
). In particular, the manifolds of Fig. 9c cover a much thicker radial domain than those of the initial ``pure bar'' model (Fig. 9a), and they also exhibit an azimuthal deformation following from the non-uniform distribution of mass shown in Fig. 9b.
A different methodology to produce response models of barred-spiral galaxies has been proposed by Patsis (2006). As already emphasized in Tsoutsis et al. (2008), the theory of the invariant manifolds explains many features of such response models. Patsis (2006) considers an ensemble of particles with initial conditions on circular orbits of the axisymmetric part of the potential, placed uniformly on the disk at radii corresponding to Jacobi constants up to the L4 value of the full potential. Then, by softly introducing the non-axisymmetric part of the potential (the transition time is a few pattern periods), the particles' distribution changes due to both the adiabatic change of the potential and phase mixing. As a result, the particles finally settle to orbits supporting both the bar and the spiral arms.
Inside the bar, the particles of such response models are captured
mostly in stable resonances belonging to a branch or bifurcation of
the x1 family, which is the continuation of the family of
circular orbits of the axisymmetric potential on which all the
initial conditions lie. However, near corotation and beyond chaos is
prominent. In this case the theory of the invariant manifolds
explains the capture of the particles in orbits supporting the
spiral arms. In our specific models, one can see this effect by
taking initial conditions
on the surfaces of
section such as those of Figs. 4a and 5a such that these initial
conditions a) belong to a chaotic domain of their respective
surfaces of section, and b) determine circular orbits when all the
non-axisymmetric part of the potential is turned off (the monopole
contribution of the bar, obtained by averaging radial forces with
respect to all possible azimuths, is taken into account in this
calculation). The locus of all these initial conditions is a
straight line
on the surface of section for a given
Jacobi constant. In practice, we take a narrow zone of some width
around such a line, i.e., allow also for a small value of the
epicyclic action around the circular orbits of the axisymmetric
model.
Figure 11 shows the result of running these initial conditions in
the case of model A, with the non-axisymmetric part of the potential
being turned on from the start, and for one iteration (Figs. 11a,b,
corresponding to 1.2 pattern rotation periods), or ten iterations
(Figs. 11c,d, corresponding to 12 pattern rotation periods), for the
value of the Jacobi constant
.
The initial
conditions correspond to a zone in Fig. 11a between the two gray
horizontal lines, that is we take 10 000 points uniformly distributed
within the intersection of the zone with the permissible domain of
motion. The first iterates of these points are shown with dark thick
points in the same figure, while the large number of small dots
illustrate the overall structure of the phase portrait at the chosen
value of
.
The corresponding figure in configuration plane is
shown in Fig. 11b. An obvious conclusion from Figs. 11a,b is that,
already after one iteration, the initially straight zone of initial
conditions in the surface of section is deformed so as to closely
follow a pattern induced essentially by the invariant manifolds
(compare Figs. 11a,b with Figs. 4a,b). This
phenomenon is repeated at subsequent iterations, so that after ten
iterations (Figs. 11c,d) the set of all points describes a pattern
nearly coinciding with that of the manifolds
,
but over a larger length of the latter. This picture does not change
qualitatively if the zone of initial conditions is taken to have a
width four times as large as in Fig. 11a. The resulting response of
the orbits in this case is shown in Fig. 11e, and the association of
this with the dynamics of the invariant manifolds is still quite
clear.
This behavior follows from a well known ``mixing property'' of chaotic dynamical systems, namely that any small and compact ensemble of initial conditions embedded in a large chaotic domain is deformed in subsequent Poincaré mappings so as to follow the unstable invariant manifolds of the main families of unstable periodic orbits located in the chaotic domain, while preserving its measure in the same domain (see Contopoulos & Harsoula 2008, and references therein for a detailed exploration of this phenomenon in the case of the ``standard map''). In the case of barred galaxies, this type of response of the chaotic orbits to the invariant manifolds generates patterns such as those of Figs. 11b,d i.e., spiral arms.
It should be stressed, however, that Figs. 11b to 11e were calculated
with the full non-axisymmetric potential, i.e.,
,
turned on from the start, i.e., without some
initial transient time of growth of the non-axisymmetric part, and
for just one value of the Jacobi constant. Figure 11f shows a
``response model'' obtained precisely via the method suggested by
Patsis (2006). We thus take the initial conditions of 105particles on circular orbits of the axisymmetric part of the
potential and on part of a uniformly populated disk, with distances
in a range corresponding to the whole range of Jacobi constant
values up to
of the full potential. The
non-axisymmetric part of the potential was now introduced softly (in
two pattern periods), and the particles were found to settle to a
nearly invariant distribution in the configuration space after about
15 pattern periods. Figure 11f shows this distribution at a time
corresponding to the 25th pattern period. Clearly, the particles
have settled to orbits supporting both the bar and the imposed
spiral arms. But the most interesting feature of this distribution
is that the maxima of the density of the response spirals depart
from the maxima of the imposed spiral at nearly the same points
(points A and A') where this happens for the invariant manifolds
of Fig. 4b, which are also essentially traced by the points in
Figs. 11b,d, or e. The interest of this result lies in that the
initial conditions of the response model of Fig. 11f are selected
from a nearly homogeneous distribution in space, and thus they are
by no means associated with the invariant manifolds. In this
respect, the invariant manifolds play for chaotic orbits a role
similar to that of stable resonances for regular orbits, i.e. the
manifolds are able to capture the chaotic orbits in their
neighborhood and to create response spiral arms.
4 Conclusions
In the present paper we examined the applicability of the theory of the invariant manifolds emanating from the unstable short period orbits around the Lagrangian points L1 and L2 of a barred galaxy in the self-consistent models of Kaufmann & Contopoulos 1996) which are rough models representing some features of real barred-spiral galaxies. We also tested the theory in weak or mean spiral versions of these models. Our conclusions are the following:
- 1.
- When both the bar and spiral components of the self-consistent
models are taken into account, the projection of the invariant
manifolds on the configuration space produces a pattern that follows
closely the imposed spiral pattern of the self-consistent model.
- 2.
- The addition of the spiral potential produces two effects: a) it
shifts the position of the unstable Lagrangian points L1, or
L2, both radially and angularly, and b) it enhances chaos
locally, in a zone around corotation, due to the increase of the
amplitude of the non-axisymmetric perturbation. The azimuthal
displacement of the Lagrangian points is the most important
factor for the manifolds to obtain a spiral form. In artificial
models in which the whole non-axisymmetric perturbation is `aligned'
to the bar (e.g. by setting the phase of the spiral term equal to
), the manifolds yield ring rather than spiral structures. If the amplitude of the perturbation is pushed to very high values, the lowest order lobes of the invariant manifolds can determine spiral patterns. However, we find that such patterns i) do not fit the spiral patterns of the self-consistent models; and ii) disappear after a few iterations of the calculation of the manifolds. This could probably be improved also by considering a lower n-value of the Ferrers bar, yielding a more gradual decrease of the bar force beyond the bar's end, and/or rendering the bar's outline more rectangular.
- 3.
- We construct a simple theoretical model yielding the
boundaries of the apocenters of the chaotic orbits in the
configuration space. Since the manifolds are developed within these
boundaries, we demonstrate that they necessarily take the form of
spiral arms, provided that the azimuths of the Lagrangian points
L1, L2 are shifted with respect to the bar's major axis. The
outermost radial limits of the manifolds are posed by rotational KAM
curves outside corotation. However, in practice there are more
stringent limits posed by the existence of cantori limiting the
chaotic diffusion. The cantori define a narrow stickiness zone
beyond corotation. The stickiness causes confinement of the
invariant manifolds inside this zone for times comparable to the
age of the galaxy.
- 4.
- A simple iterative method is proposed to calculate ``spiral response'' models on the basis of the theory of the invariant manifolds, starting from the invariant manifolds of a pure bar model. We also discuss the relevance of the response models of Patsis (2006) to the theory of the invariant manifolds. Following a general property of the Hamiltonian flow in a connected chaotic domain, a small sub-domain of initial conditions embedded within the chaotic domain yields successive Poincaré maps following closely the form of the invariant manifolds of the main families of unstable periodic orbits in the same domain. The consequences of this effect are tested by calculating a full response model via the method proposed by Patsis (2006). Despite the fact that the initial conditions of the particles in the latter model have no relevance to the invariant manifolds, the maxima of the response spiral arms in the final state are found to follow closely the underlying pattern formed by the invariant manifolds.
Acknowledgements
P. Tsoutsis and C. Kalapotharakos were supported in part by the Research Committee of the Academy of Athens. We thank an anonymous referee for the numerous comments which improved the paper.
Appendix A: Permissible regions for motion
The Hamiltonian (1), where
and
defines a curve of zero velocity
given by
![]() |
(A.1) |
where V0(r) and






![]() |
(A.2) |
where


Consider a very simple form for V1, e.g.
.
For all values of the Jacobi constant between
and
we have h-A<0, thus, since
is small, we
find from (A.2) that M>0, or
.
Omitting terms
of third order in
,
we find from Eq. (A.2)
![]() |
(A.3) |
If -A<h<A the motions are outside two banana-like curves around L4 and L5. In particular, for



![]() |
(A.4) |
Equation (A.4) gives the intersections of the banana-like curves of zero velocity on the axis connecting the center to the stable Lagrangian points. The forbidden regions of the CZV can be compared with the corresponding forbidden regions of the surface of section





![]() |
(A.5) |
This gives
![]() |
(A.6) |
Since

References
- Aguerri, J. A. L., Hunter, J. H., Prieto, M., et al. 2001, A&A, 373, 786 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
- Buta, R., Vasylyev, S., Salo, H., & Laurikainen, E. 2005, AJ, 130, 506 [NASA ADS] [CrossRef] (In the text)
- Contopoulos, G. 1981, A&A, 102, 265 [NASA ADS] (In the text)
- Contopoulos, G. 2002, Order and Chaos in Dynamical Astronomy (Berlin: Springer-Verlag) (In the text)
- Contopoulos, G., & Grosbøl, P. 1988, A&A, 197, 83 [NASA ADS] (In the text)
- Contopoulos, G., & Harsoula, M. 2008, Int. J. Bif. Chaos, 18, 1 [CrossRef] (In the text)
- Contopoulos, G., Harsoula, M., Voglis, N., & Dvorak, R. 1999, J. Phys. A: Math. Gen., 32, 5213 [CrossRef] (In the text)
- Efthymiopoulos, C., Contopoulos, G., Voglis, N., & Dvorak, R. 1997, J. Phys. A: Math. Gen., 30, 8167 [NASA ADS] [CrossRef]
- Fux, R. 2001, A&A, 373, 511 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
- Grobman, D. M. 1959, Dokl. Akad. Nauk SSSR, 128, 880 (In the text)
- Grosbøl, P. 2003, in Galaxies and Chaos, ed. G. Contopoulos, & N. Voglis, Lect. Notes Phys., 626, 201 (In the text)
- Hammed, S., & Devereux, N. 1999, ApJ, 118, 730 (In the text)
- Hartman, P. 1960, Proc. Amer. Math. Soc., 11, 610 [CrossRef] (In the text)
- Kaufmann, D. E., & Contopoulos, G. 1996, A&A, 309, 381 [NASA ADS] (In the text)
- Kaufmann, D. E., & Patsis, P. 2005, ApJ, 624, 693 [NASA ADS] [CrossRef] (In the text)
- Laurikainen, E., & Salo, H. 2002, MNRAS, 337, 1118 [NASA ADS] [CrossRef] (In the text)
- Lindblad, P. A. B., Lindblad, P. O., & Athanassoula, E. 1996, A&A, 313, 65 [NASA ADS] (In the text)
- Patsis, P. A. 2006, MNRAS, 369, 56 [NASA ADS] (In the text)
- Patsis, P. A., Efthymiopoulos, C., Contopoulos, G., & Voglis, N. 1997, A&A, 326, 493 (In the text)
- Pfenniger, D. 1984, A&A, 134, 373 [NASA ADS] (In the text)
- Pfenniger, D., & Friedli, D. 1991, A&A, 252, 75 [NASA ADS] (In the text)
- Pichardo, B., Martos, M., & Moreno, E. 2004, ApJ, 609, 144 [NASA ADS] [CrossRef] (In the text)
- Romero-Gomez, M., Masdemont, J. J., Athanassoula, E. M., & Garcia-Gomez, C. 2006, A&A, 453, 39 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
- Romero-Gomez, M., Athanassoula, E. M., Masdemont, J. J., & Garcia-Gomez, C. 2007, A&A, 472, 63 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
- Schwarzschild, M. 1979, ApJ, 232, 236 [NASA ADS] [CrossRef] (In the text)
- Smith, B., Miller, R. 1982, ApJ, 257, 103 [NASA ADS] [CrossRef] (In the text)
- Sparke, L. S., & Sellwood, J. A. 1987, MNRAS, 225, 653 [NASA ADS] (In the text)
- Tsoutsis, P., Efthymiopoulos, C., & Voglis, N. 2008, MNRAS, 387, 1264 [NASA ADS] [CrossRef] (In the text)
- Tully, R. B., Verheijen, M. A. W., Pierce, M. J., Huang, J.-S., & Wainscoat, R. J. 1996, ApJ, 112, 2471 [NASA ADS] (In the text)
- Voglis, N., Tsoutsis, P., & Efthymiopoulos, C. 2006a, MNRAS, 373, 280 [NASA ADS] [CrossRef] (In the text)
- Voglis, N., Stavropoulos, I., & Kalapotharakos, C. 2006b, MNRAS, 372, 901 [NASA ADS] [CrossRef]
All Tables
Table 1:
Parameters of models A, B, C (from Kaufmann & Contopoulos
1996). The units are
for
A,
for
,
,
and
,
for r1, r2,
,
a, b, c, and
,
for
,
,
for
,
and
/pc2 for
.
Model A' has the same
parameters as model A, except for the spiral amplitude A=1000 and
the pitch angle
.
Model B' has the same parameters
as model B except for A=2500,
,
,
.
All Figures
![]() |
Figure 1:
a) Projections of the invariant manifolds
|
Open with DEXTER | |
In the text |
![]() |
Figure 2: The ratio of non-axisymmetric forces, due to the bar or to the spiral arms, versus total axisymmetric force, as a function of the distance R along the x-axis (solid) or y-axis (dashed) for models A, B, and C (panels a), b), and c) respectively). The vertical dashed lines mark the distance of the L1 or L2points in each case. |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
The |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
a) Phase portrait near corotation (
|
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Same as in Fig. 4, but for model B. The Jacobi constant
is
|
Open with DEXTER | |
In the text |
![]() |
Figure 6:
a), b) Same as in Figs. 4a,b but for the model A', and
|
Open with DEXTER | |
In the text |
![]() |
Figure 7:
The curves of zero velocity (equipotential curves of
the effective potential
|
Open with DEXTER | |
In the text |
![]() |
Figure 8:
The resonant phase space structure in the corotation
region in the cases of a) model A with
|
Open with DEXTER | |
In the text |
![]() |
Figure 9:
A ``spiral response'' model based an initially ``pure
bar'' version of model A. a) Manifolds of the pure bar case for
|
Open with DEXTER | |
In the text |
![]() |
Figure 10:
Unstable invariant manifolds of the PL1 and PL2 orbits in a
pure bar version of model A in which the bar's mass and pattern
speed are altered with respect to the reference values
|
Open with DEXTER | |
In the text |
![]() |
Figure 11:
Model A for the value of the Jacobi constant
|
Open with DEXTER | |
In the text |
Copyright ESO 2009
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