Mon. Not. R. Astron. Soc. 369, L56–L60 (2006)
doi:10.1111/j.1745-3933.2006.00174.x
The stellar dynamics of spiral arms in barred spiral galaxies
P. A. Patsis
Research Centre for Astronomy, Academy of Athens, Soranou Efessiou 4, GR-11527, Athens, Greece
Accepted 2006 March 30. Received 2006 March 28; in original form 2006 March 15
ABSTRACT
A dynamical mechanism is proposed that explains the spiral structure observed frequently as a
continuation of the bars in barred spiral galaxies. It is argued that the part of the spirals attached
to the bar is due to chaotic orbits. These are chaotic orbits that exhibit for long time intervals
a 4:1-resonance orbital behaviour. They are of the same type of orbit as is responsible for
the boxiness of the outer isophotes of the bar in cases like NGC 4314, as indicated by Patsis,
Athanassoula & Quillen. The spirals formed this way are faint with respect to the bar, open as
they wind out, and do not extend over an angle larger than π/2. A possible continuation of the
spiral structure towards larger angles can be due to orbits trapped around stable periodic orbits
at the corotation region. We present a family of stable, banana-like periodic orbits, precessing
as E J increases, that can play this role.
Key words: stellar dynamics – galaxies: kinematics and dynamics – galaxies: spiral.
1 INTRODUCTION
The spiral structure of disc galaxies does not have a unique morphology. Even grand-design spirals differ in their pitch angles, their
range of extent and amplitude variation along the arms. In some
cases of barred spiral galaxies, the spirals seem to emerge out of
the bar as a continuation beyond its apocentres, which are located
along the major axis. A typical example is NGC 4314. Optical observations of this galaxy can be found in Knapen et al. (2004), while
near-infrared images can be found in Quillen, Frogel & Gonzalez
(1994). In NGC 4314 the spirals are faint with respect to the bar and
open as they wind out. The arms do not extend over an angle larger
than π/2. However, in other cases of spirals emerging out of the ends
of bars the arms are stronger and complete almost a 2π angle without forming a ring. As a typical example we mention NGC 3513.
There are also barred galaxies where a quadruple arm structure is
present. In spiral structures of this type it is usually difficult to trace
the origin of the lower set of arms, the one closer to the galactic
centre, on the major axis of the bar. As an example we mention the
case of NGC 1672. Enlightening images of the two latter galaxies
can be found in Sandage (1988), in panels 52 and 39 respectively.
The present Letter concerns the dynamics of this kind of spiral.1
In Section 2 we describe briefly the model used in the calculations,
in Section 3 we present the results obtained, and in Section 4 we
discuss our conclusions.
E-mail: [email protected]
1
We note that their morphology is quite different from that of almost perfectly logarithmic spirals encountered in normal (i.e. weakly barred, or
non-barred) spirals such as e.g. in NGC 3223 (fig. 4 in Grosbøl & Patsis
1998).
2 THE MODEL
In Patsis, Athanassoula & Quillen (1997, hereafter PAQ), the potential estimated by Quillen et al. (1994) for NGC 4314 has been
used to study the orbits in the bar of this galaxy, which is a typical
early-type bar. One of the basic conclusions of that study was that
the characteristic outer boxy isophotes are due to chaotic orbits.
Stable periodic orbits with boxy morphology exist, but influence a
tiny space in phase-space.
As we move from the centre of NGC 4314 outwards, the boxy
isophotes become more and more pointed and finally form the beginning of a spiral structure (fig. 1 in Quillen et al. 1994). This is the
basic morphology studied in this Letter, thus the potential in Quillen
et al. (1994) is an ideal candidate to be used for the dynamics of this
and similar galaxies.
The potential has been calculated using near-infrared observations in the J, H and K bands under the assumption of a constant
mass-to-light (M/L) ratio. In the z = 0 plane it can be approximated
by the function
(r , θ ) = 0 (r ) +
mc (r ) cos(mθ) + ms (r ) sin(mθ ),
(1)
m>0
where m = 2, 4, 6.
The amplitudes ofthe various components in equation (1) are
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written in the form
α r n . The values of the coefficients α n
n=0 n
−1 2
in (km s ) can be found in table 1 of Quillen et al. (1994). Recently Byrd, Freeman & Buta (2006) provided evidence that the
M/L ratio varies in barred galaxies like NGC 3081. Even if this
is the case also for NGC 4314, the potential that we use is qualitatively correct for a typical barred galaxy with a weak spiral
structure.
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Equations of motion are derived from the Hamiltonian
1 2
1
(ẋ + ẏ 2 ) + (x, y) − 2b (x 2 + y 2 ) = E J ,
(2)
2
2
where (x, y) are the coordinates in a Cartesian frame of reference
coroating with the bar with angular velocity b . (x, y) is the
potential in Cartesian coordinates, E J is the numerical value of the
Jacobian integral and dots denote time derivatives. Throughout this
Letter E J is given in (km s−1 )2 . The presentation of the results is
given in frames, where the major axis of the bar is along the vertical,
y-axis.
In PAQ a particular value of b (44.96 km s−1 kpc−1 ) has been
used in order to match in the best way the size of the observed bar
with the orbits found. However, it has been realized that the basic
dynamics of the model did not change for a range of b values
that give a corotation radius r C , that is 3.8 < r C < 4.5 kpc. In the
present study we have examined bars rotating with 38.23 b 46 km s−1 kpc−1 . There was no difference as regards the overall dynamics found. For the presentation of the results here the value
b = 38.23 km s−1 kpc−1 is used. For this value the unstable
Lagrangian points L 1,2 are located at a radius r ≈ 4.5 kpc. It should
be noted that the potential minima bend and deviate from the major
axis of the bar beyond a radius of about 4 kpc.
H≡
3 R E S U LT S
We first calculated a number of response models of stellar discs
under the influence of the potential in Equation (1), rotating with
b . Fig. 1 gives the result after 10 pattern rotations of a typical case.
It is an image of the snapshot on which a smoothing filter has been
applied. Darker areas correspond to larger local surface densities and
a logarithmic scale is used. 20 000 particles have been distributed
initially randomly on a disc of 6-kpc radius with constant surface
density and have been put in circular motion. In a preliminary phase,
the amplitude of the bar has been grown from 0 to its maximum
value within three pattern rotations, so that the particles do not feel
the perturbation abruptly. Particles that spent time corresponding to
more than one bar period at distances larger than 8 kpc have been
Figure 1. A typical stellar response model after 10 pattern rotations. b =
38.23 km s−1 kpc−1 . Characteristic morphological features of NGC 4314,
such as the boxy isophotes at the end of the bar and the weak spiral structure
beyond the end of the bar, are reproduced as indicated by the overplotted
isodensities. The bar rotates counterclockwise.
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removed and replaced by an equal number of particles so that we
have followed the response of a constant number of particles until
the end of the run. The new particles have been distributed again
randomly, in circular motion. The potential is not accurate at large
distances, so one does not want to spend time integrating the orbits
there.
The result is typical for a large number of response models that
differ in their initial conditions (positions and velocities), pattern
speeds, time within which the perturbation reaches its full strength
and replenishment or not of the particles that reach large distances
from the centre. The differences are small, refer mainly to the early
snapshots and will not be discussed here. Particularly useful for
understanding the dynamics of the system were models where the
initial conditions have been distributed only at some parts of the
disc. So we have followed the evolution of particles with initial
conditions only in a rectangle of width 1 kpc and length 12 kpc
(from −6 to 6 kpc) along the minor axis, or in squares of side 1 kpc
and centred on the unstable Lagrangian points L 1 and/or L 2 . The
snapshots of a model look very similar after the first three pattern
rotations. It is remarkable how fast two morphological features that
characterize NGC 4314 are developed, namely the boxiness of the
outer isodensities of the bar and the weak spiral structure, which
fades out away from the ends of the bar.It is evident that the particles
in the simulation follow similar orbits like the orbits of the stars in
the galaxy. This offers a big advantage for dynamical studies, since
one can easily trace the regions of the phase space that are populated.
It also tells us that a bar perturbation in the potential growing over a
few pattern periods leads the particles to trajectories that reproduce
the morphology of the galaxy.
We have examined the orbital content of the spirals by calculating statistics on the E J values of the particles located at radii r >
4 kpc. Most of these particles are in the spiral region. The histogram
corresponding to the snapshot of the model in Fig. 1 is given in
Fig. 2. It is obvious that the peak of the histogram is close to the E J
value of the unstable Lagrangian points (E L1,2 = −47 806.2). However, there are a substantial number of particles at larger energies.
For all models we have followed the time evolution of the particles and the corresponding velocity fields in successive snapshots.
In parallel, for all models, we have considered surfaces of section at the critical region −50 000 < E J < −40 000 with a step
E J = 250. Since we are close to corotation most orbits are, as
expected, chaotic. In order to identify orbital patterns, which are
possibly followed by the particles in the response models, a number
of orbits on the surfaces of sections have been integrated for times
Figure 2. A histogram giving the distribution of the E J values of the particles that support the spiral structure in Fig. 1. In this model we have E L1,2 =
−47 806.2.
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P. A. Patsis
corresponding to six pattern rotations. For this purpose we have used
initial conditions on dense grids covering the surfaces of section and
we have created a library of orbits (integrated initial conditions) that
can be encountered at each E J for an integration time corresponding
to six rotations of the bar. Besides getting the information about the
stability of each orbit by simple inspection, this has allowed us also
to calculate statistics about the regions on the surfaces of sections
that are visited by the orbits in the response models. For this we
have just integrated backwards in time specific particles at the end
of a simulation. Finally we have followed the characteristics and
the evolution of the stability index of the families of periodic orbits
at the corotation region for all models with different b that we
studied.
It has been realized that about 80 per cent of the integrated orbits on the surfaces of section for −49 000 < E J < −46 000 are
chaotic orbits that for several pattern rotations are of rhomboid- or
rectangle-like morphology. The particles on the spirals are on orbits of this kind that at a certain time pass close to the unstable
Lagrangian points and then describe large arcs away from the bar
contributing to the spiral structure. Some of them visit the region
of both unstable Lagrangian points a few times before departing
from the bar region. However, the vast majority of them do not visit
the corotation region again in order possibly to reinforce the spiral
arms. Models with initial conditions in boxes around the unstable
Lagrangian points and without replenishment of the ‘lost’ particles
are depleted at the end of the runs. Despite the fact that the potential used is not accurate at large distances and steep gradients
could lead the particles quickly to large distances, the weakness of
the spirals away from the bar observed in NGC 4314 indicates that
the effect described with this set of response models describes the
real situation in NGC 4314. Arcs of the spiral arms that help
the continuation of the spirals at angles away from the apocentres of the bar are more easily created in models with random initial
velocities, or with initial conditions in a narrow strip of width 1 kpc
along the x-axis.
In Fig. 3, we illustrate what we discuss at this point, in a model
with initially 20 000 particles put in circular motion in a box centred
on L 1 . The particles that stay away from the 6-kpc disc for one period
of the bar are not replaced. The overplotted trajectory shows the orbit
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3
2
1
0
-1
-1
0
1
2
3
4
5
6
7
Figure 4. The velocity field along one of the arms in another snapshot of
the same model as in Fig. 3. We observe that the flow is roughly parallel to
the spiral.
of a particle located on the arms after eight pattern rotations in the
past (blue curve) as well as in the future (magenta curve).
It is worth noting that in all snapshots of the models studied, the
flow along the spirals is smooth, despite the fact that the orbits are
chaotic. The particles seem to be ejected from the end of the bar in
ballistic trajectories. We show this in a zoom into one of the arms
in the velocity field of another snapshot of the model in Fig. 3. This
is given in Fig. 4.
Until now we have discussed the orbits that make the spiral and
have, for b = 38.23 km s−1 kpc−1 , −49 000 < E J < −46 000. For
E J > −47 000 we have theoretically the presence of banana-like
orbits in the models. They are mostly unstable. In Fig. 5 is given
the projection of the (E J , x) characteristic of several families of this
kind. The black curve at the upper left corner is the curve of zero
velocity for y = 0. The only stable regions on the characteristics are
tiny segments at the turning points of the curves, a small branch of
a multiplicity 2 family close to E J ≈ −45 700, and along the lower
part of the curve to the right of point ‘A’. This characteristic belongs
to a family of banana-like orbits, precessing as E J increases. We
note that the main family of banana-like short-period orbits (SPOs)
(Contopoulos 2002) is not depicted here, since it never cuts the
y = 0 axis. SPO orbits leaving the stable Lagrangian points outside
have been presented by Contopoulos (1978). The question is what
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0
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-2
0
2
4
6
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Figure 5. (E J , x) projection of the characteristics of the main banana-like
families in the models with model b = 38.23 km s−1 kpc−1 . They are
mainly unstable. The black curve at the upper left corner is the curve of zero
velocity for y = 0.
Figure 3. The orbit of a particle located at the arm region after eight pattern
rotations. Its orbit in the past is plotted in blue, while its journey in the future
is plotted in magenta. The orbit is overplotted on the velocity field of a model
with initial conditions only in a square of side 1 kpc at the L 1 region.
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Figure 6. Chaotic banana-like orbits for −46 000 < E J < −42 000. Under certain conditions they can also support the spiral arms.
help do the banana-like orbits associated with the L 4,5 Lagrangian
points provide to the spiral structure if they are populated.
Typical chaotic orbits for −46 000 < E J < −42 000 are given in
Fig. 6. The form of these orbits is described in Contopoulos (2002)
(section 3.1.7). In our simulations we encountered such orbits in
models with particle replenishment and random initial velocities,
and in models where the initial conditions were distributed in a
narrow ribbon-like arrangement around the x-axis. In all these cases
their role was auxiliary and had as the main effect the reinforcement
of the inner part of the spiral, giving a double-arm character close
to either end of the bar.
The presence of quasi-periodic orbits, trapped by the orbits of the
stable family, which precess as E J increases, strengthens the spirals
in angles away from the unstable Lagrangian point at which they
emerge. Their role in the reinforcement of the spiral is auxiliary
as well. They contribute to the effect of a bifurcation of the arms,
which exists in general in all models, but is much weaker without them. In this category of orbits one has to count (besides the
quasi-periodic orbits) sticky orbits around them, as well as chaotic
orbits at the same E J , which could support the arms in the same
way as chaotic orbits support the outer boxy isophotes of the NGC
4314 bar in PAQ. The effect of these orbits on the dynamics of
the spirals is described in Fig. 7. The snapshot is of a model with
random initial velocities and an abrupt introduction of the perturbation term in the potential. With respect to other models, in this case
we see in the velocity fields an increasing number of particles that
emerge out of the bar at distances closer to the minor axis of the
bar.
4 DISCUSSION AND CONCLUSIONS
This Letter refers to disc galaxies that have a bar and a spiral
and both components share the same pattern speed. Kaufmann &
Contopoulos (1996) were the first to point out that in such systems
chaotic orbits support the inner portions of their spiral arms. According to them, at distances beyond the outer 4:1 resonance, the
spirals are supported by regular orbits trapped around the central
family. Here it is proposed that the chaotic orbits that support spiral
arm arcs attached to the ends of the bar are chaotic orbits that for
long time intervals support inner ‘4:1’ resonance structures and the
boxiness of the outer bar isodensities. At a certain point these orbits
find a path through the region of the unstable Lagrangian points and
‘escape’ to large distances away from the bar. Our calculations have
used the potential estimated by Quillen et al. (1994) for the galaxy
NGC 4314. The remarkable similarity between the morphology of
this galaxy and the stellar response models for all different initial
conditions used underlines the fact that the orbital dynamics at the
outer parts of the model and the galaxy are the same.
In the case of NGC 4314, the spiral structure ends before it completes a π/2 angle away from the ends of the bar, as happens in
general in our response models. However, there are cases of other
barred galaxies where the spiral arms continue. In such cases, either
the Kaufmann–Contopoulos scenario applies, or the spiral is helped
to remain strong by chaotic banana-like orbits, and/or quasi-periodic
orbits trapped around the family of stable periodic orbits depicted in
Fig. 7(b). The case of NGC 1672 is a reasonable candidate to have
a spiral structure supported by the latter two types of orbits.
Figure 7. (a) The velocity field after eight pattern rotations in a model with initially random velocities and an abrupt introduction of the perturbation term in
the potential. (b) Stable periodic orbits of a banana-like family, which precess as E J increases, overplotted on the velocity field show how they can strengthen
the spiral arms at large angles away from the end of the bar from which they emerge.
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P. A. Patsis
Recently, Voglis and collaborators (private communication) have
suggested that the whole spiral structure is due to chaotic orbits related to the invariant manifolds of the unstable Lagrangian points,
while Romero-Gomez et al. (2006) have indicated a similar mechanism for the formation of rings.
We note that spirals attached to the ends of the bars are difficult
to observe in stellar response models, when the imposed potential
is symmetric (fig. 12 in Contopoulos & Patsis 2006), while they are
present in the gaseous responses under the same potential (fig. 13
in Contopoulos & Patsis 2006). Important for the appearance of
these spiral features in the case of the NGC 4314 potential is the
displacement of the unstable Lagrangian points from the major axis
of the bar.
The mechanism that we discuss refers to systems with one pattern
speed. If the bar and spirals do not share a common pattern speed,
the orbits due to the bar potential do not support the extended spiral
structure, as is shown in the case of NGC 3359 (Boonyasait, Patsis
& Gottesman 2005). Our conclusions are summarized as follows.
AC K N OW L E D G M E N T S
This work was partly supported by the Research Committee of the
Academy of Athens. It is a pleasure to acknowledge fruitful discussions about bar dynamics with G. Contopoulos, E. Athanassoula,
N. Voglis, P. Grosbøl and D. Kaufmann. I also thank the referee G.
Byrd for helpful comments that improved the paper.
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(i) The spiral structure that appears attached to the end of a bar
is due to chaotic orbits of the ‘4:1 resonance type’, which for long
times support the boxiness of the outer isodensities of the bar. The
orbital content of the two features (weak spirals and boxiness) is the
same as that described in PAQ.
(ii) The flow of stars along the spirals in the systems that we
study is smooth, as given in Fig. 4.
(iii) Other chaotic orbits in banana-like motion for long times
reinforce a double-arm character close to the end of the bars.
(iv) There is a stable family of banana-like orbits precessing at
successive E J that, if populated, helps the spiral structure away from
the ends of the bar. If gas follows similar paths, then at these parts of
the spirals star formation would be favoured because of streaming
motions. These orbits may also support the appearance of isolated
spiral fragments.
This paper has been typeset from a TEX/LATEX file prepared by the author.
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