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-- Main.MerceRomero - 15 Jan 2009

Here we consider the analytic potential to describe the Galactic bar and the axisymmetric component as in Pichardo et al. The procedure followed here to compute the invariant manifolds is described in detailed in  [[http://www.aanda.org/index.php?option=article&access=bibcode&bibcode=2006A%2526A...453...39RPDF][Paper I]], and applied for several bar potentials in [[http://www.aanda.org/index.php?option=article&access=bibcode&bibcode=2007A%2526A...472...63RPDF][Paper II]]. Briefly, we locate the equilibrium points on the galactic plane, we study their linear stability. We then concentrate on the two saddle points, L1 and L2, located at the ends of the bar. In all the plots shown in the papers and here, the system corotates with the bar, so the bar is fixed along the x-axis. We compute the family of periodic orbits around the eqilibrium points and for each periodic orbit (i.e., for each energy - Jacobi constant), we compute the invariant manifolds associated to the periodic orbit. In "easy words", invariant manifolds are asymptotic orbits that depart/tend to the periodic orbit and drive the motion of the particles around them. They are like flux tubes. They do not depart in any direction but in the two related to the saddle behaviour of the equilibrium point. So stars will follow orbits linked to these manifolds. 

We first analyzed what parameters of the model have an influence in the outer part of the Galaxy by computing the effective potential along the x-axis. Here we found that only the Bar mass and its pattern speed show different curves in the outer parts. So the next step is creating a grid where in the x-axis we decrease the pattern speed and in the y-axis we increase the Bar mass, the rest of the parameters are fixed as in the standard model in B. Pichardo's thesis. For each model, we perform the study described above and the result is in the figure attached below, plot2dbar.pdf. Similar plots can be found in Paper II for three other bar models.

The coding for the colours is the following: black for spirals, orange for rR1 pseudorings, green for rR1 rings and magenta for rings that maybe considered as single circular outer rings. The red lines delineate the locus of the spiral arms. There is a different trend in this figure compared to the figures in [[http://fr.arxiv.org/abs/0807.3832][Paper III]] . So here I'd like to study in detail these models and check what happens there.

In the next step I proceeded the same way as with the bar potential. Here I consider the potential resulting from the superposition of the axisymmetric component and the spiral arms. I analysed each component of the model to check which one has an effect to the outer parts of the galaxy and, again, the spiral mass and its pattern speed are the two parameters with a major influence. I make families of models by fixing the rest of the parameters and varying "alfa2b" and "omegs" within a range of values. For each model, I compute the invariant manifolds and I plot the resulting morphology in a 2D grid. Analogously to that of the bar potential, I decrease on the x-axis the spiral pattern speed from left to right and, in the y-axis I decrease the spiral mass from top to bottom. The results are in the file: plot2dsp.pdf, attached below. Again, the locus of the spiral is plotted in red lines. There are some models, were self-consistency is achieved.

The next step is performing the same study when we consider the axisymmetric + bar + spiral, i.e. the total potential, with the bar and spirals rotating at the same pattern speed. The results are shown in the next two plots of the table below. In the first case, I fixed the value of alfa2b=0.05 and I vary the bar mass and the pattern speed (plot2dbarsp05.pdf). In the second case, I increase the spiral mass to alfa2b=0.1 (plot2dbarsp1.pdf). The plot plot2dbar.pdf represents the case when alfa2b=0. 

In this last case, we obtain some nice self-consistent spiral arms corresponding to realistic values for the bar mass and the pattern speed. So we will study them in detail.

Finally, I change the bar model to that used by Dehnen in his paper of 2000 AJ 119, 800-812: "The effect of the OLR of the galactic bar on the local stellar velocity distribution". So I take the power-law to model the axisymmetric component and the cos(2theta) potential of the bar. There I include another free parameter, namely the exponent in the A(r) function. He sets it to 3, and I will change it in other cases (as I do in our papers with Lia). The last figure in the table is the 2D grid for a flat rotation curve (beta=0) and the same potential he uses (expb=3). There I vary the bar strength characterized by the parameter alpha (close related to the amplitude of the bar) in the same range he uses in the paper, and in the x-axis, I vary the pattern speed as usual, also in the same ranges as he uses. Note that here the model is the superposition of an axisymmetric component plus a bar. So NO spiral potential is included. The results are as expected. No clear spiral arms are present. And the shape of the manifolds delineate approximately the shape of the OLR closed orbits, with an R1 shape. 

Now I'm planning to change the shape of the rotation curve (by varying beta) and the shape of the force curve (by varying expb). If we make beta lower than 0, the rotation curve falls steeply at the outer parts. The results in the case of a falling rotation curve and the exponent 3 in the model are in the plot "plot2Dbeta-02exp3.pdf", below. As expected, when we have a falling rotation curve, more spiral cases appear at the upper left corner of the plot, but they do not expand up to the solar radius (solid circle).

I also include in each panel a circle of radius 8.5 to show the solar radius, and as before, the locus of the spiral.

Attachments

Topic attachments
I	Attachment	History	Action	Size	Date	Who	Comment
pdf	Romero.pdf	r1	manage	6974.5 K	2009-03-13 - 14:17	MerceRomero	Talk at a conference, with a summary of the dynamics
pdf	plot2Dbeta-02exp3.pdf	r1	manage	9082.8 K	2009-06-25 - 09:07	MerceRomero	2D grid varying the pattern speed and bar strength (alpha). Beta=-0.2 (falling rotation curve) and exp=3. Dehnen00 model
pdf	plot2Dbeta0exp3.pdf	r1	manage	8338.2 K	2009-06-23 - 09:59	MerceRomero	2D grid varying the pattern speed and bar strength (alpha). Beta=0 and exp=3. Dehnen00 model
pdf	plot2dbar.pdf	r1	manage	6019.2 K	2009-03-13 - 14:02	MerceRomero	2D grid varying the pattern speed and the Bar mass. Barbara's models
pdf	plot2dbarsp05.pdf	r1	manage	7806.7 K	2009-06-23 - 09:36	MerceRomero	2D grid varying the pattern speed and the Bar mass. Spiral mass=0.05. Barbara's models
pdf	plot2dbarsp1.pdf	r1	manage	7243.0 K	2009-06-23 - 09:40	MerceRomero	2D grid varying the pattern speed and the Bar mass. Spiral mass=0.1. Barbara's models
pdf	plot2dsp.pdf	r1	manage	7675.7 K	2009-06-23 - 09:04	MerceRomero	2D grid varyind the pattern speed and the spiral mass. Barbara's models

Topic revision: r9 - 2009-06-25 - MerceRomero