A Self-Consistent Dynamical Model for the {sl COBE} Observed

A self-consistent stellar dynamical model for the Galactic bar is constructed from about 500 numerically computed orbits with an extension of the Schwarzschild technique. The model fits the {\sl COBE} found asymmetric boxy light distribution and the observ

astro-ph/9508146 30 Aug 1995

A self-consistent stellar dynamical model for the Galactic bar is constructed from about 500 numerically computed orbits with an extension of the Schwarzschild technique. The model fits the {\sl COBE} found asymmetric boxy light distribution and the observ

Figure 1. The left and right panels show the self-consistent model from the Schwarzschild technique and the evolution of its N-body counterpart respectively. The left three panels are the predicted surface density map (solid contours, 1 magnitude spacing in between), the kinematics along b= 4 and b= 7

, and the radial dispersion ( ), the proper motions (, ) and the cross term along the minor axis. The corresponding observations are also plotted. The right panels show the N-body realization in the beginning (to the left), and the con guration after evolving for 10 rotation periods (to the right) both in the face-on view and edge-on view. The lower two panels shows the global equilibrium indicators as functions of time more quantitatively, which include the Virial equilibrium indicator 2K=W and the normalized pattern speed (both should be unity if in steady state), the three moments of inertia I, I and I along the three principal axes (should all be constant) and the rest frame cross term I (should be sinusoidal).o o r l b lr xx yy zz XY

A self-consistent stellar dynamical model for the Galactic bar is constructed from about 500 numerically computed orbits with an extension of the Schwarzschild technique. The model fits the {\sl COBE} found asymmetric boxy light distribution and the observ

shape of the observed light distribution (the dust-subtracted K band COBE map from Weiland et al. 1994 is the dotted line, and the reprojection of the volume density model from Dwek et al. 1995 is the dashed line. the model is in solid line) and the fall-o of the observed radial velocity dispersions on the minor axis (data from Terndrup et al. 1995, Sharples et al. 1990), and the proper motion dispersions at Baade's Window (from Spaenhauer et al. 1992) and the vertex deviation (from Zhao et al. 1994). Note a (probably too) simple Miyamoto-Nagai disk has been added to the bar's surface density map for direct comparison with the COBE map. The stability of the model is tested by rst converting the orbit model to an N-body model. The conversion is done by spreading 50K particles randomly in the phase of each orbit with a number in proportion to the weight of each orbit. The evolution of the N-body bar is followed with the Self-Consistent Field method code (Hernquist and Ostriker 1992) for a Gigayear. In the case with a rigid disk potential, the bar is stable for at least 1 Gyr with only small evolution of the pattern speed and the shape (Figure 1). Note the elongated bar shape in the face-on view and the boxyness in the edge-on view are similar at two epochs. The nal bar has settled down to dynamical equilibrium. This is interesting since the bar model also has a self-consistent central cusp ( r 1 85) with a mass 5-10% of the bar. We nd that the central cusp does not cause the bar to dissolve rapidly through scattering the boxy orbits. The self-consistent model can directly tell us how di erent orbit families are populated. The model bar's mass is pided between explicitly integrated direct orbits and some orbits (which I call collective-orbits) with an implicit isotropic distribution function of f= f (E ). The advantage of using this hybrid representation of the bar's phase space is to get around integrating the ill-understood chaotic orbits explicitly without missing any necessary orbit families of the bar. We nd that although the dominant orbits in the model are still the direct boxy orbits (60% in mass), which are responsible for both the boxy contours in the COBE map and the rapid rotation of the bulg

e, the rest of the mass is in the collective-orbits, which implicitly contain 2=3 chaotic orbits and 1=3 retrograde orbits. As one does not expect signi cant amount of retrograde or chaotic orbits be populated during the formation of the bar from the disk, we argue that the large fraction seen in the model poses a possible challenge to this canonical scenario. As the stars and the gas share the same potential, it is of interest to study the response of gas in the COBE potential. Binney et al. (1991) argue that the pressure force and dissipational collisions tend to beam gas clouds on non-selfintersecting closed orbits, and in particular, the shape of the non-self-intersecting x1 orbits should to zeroth order match the dynamical boundaries of the HI and CO gas clouds in the longitude-velocity plane. We have reexamined this interpretation but now with a realistic potential based on tting stellar observations. We nd a fast bar with pattern speed 60 km/s/kpc and a bar angle of 10 20 degrees has a vertical edge at l= 2 due to the nearly cusped x1 orbits (the vertical loop), which could t the parallelogram of CO (not shown, but similar to Binney et al. Figure 2). The terminal velocities (the crosses) of the noncusped x1 orbits has a rapid rise and fall o which matchs the terminal velocity of the HI map (Figure 2). Models with bigger bar angles and/or smaller patternlr: J o

A self-consistent stellar dynamical model for the Galactic bar is constructed from about 500 numerically computed orbits with an extension of the Schwarzschild technique. The model fits the {\sl COBE} found asymmetric boxy light distribution and the observ

Figure 2. overplots the HI map with the l for our best model.

v digram of the x1 orbits

speed appear to be worse in the t. The HI map was kindly made electronicly available by Harvey Liszt. A microlensing map of the bar is another by-product of the model. It is built by agging the particles in the converted N-body model as lenses or sources and\observing" them along di erent line-of-sights. We nd that the optical depth of the model at MACHO and OGLE observed elds is a few times 10 6. The typical event should last about 30 days if the lenses were all one solar. Comparing these values with the observed 20 day time scale (Alcock et al. 1995, Udalski et al. 1994), we estimate that the average mass of the lenses is 0:4M, well above the brown dwarf limit of 0:1M (other results are summarized in Zhao et al. 1995a,b). We argue that most of the lens seen in these experiments are luminous stars, which have the hope to be detected with other methods as well as with microlensing. In summary, we have built a sophisticated model for the Galactic bar that is consistent with a variety of recent observations. The model is likely to be stable and unique. More rigorous tests of stability (simulations with a live disk and halo) and exploring the full range of plausible models still remain to be done. The model results will be made electronicly available to the community in the near future. Acknowledgments. This work is a bulk part of the PhD thesis work done in Columbia University. I thank David Spergel and Michael Rich for advices and supports during this work.

DiscussionD. Pfenniger: Ho

w do you choose your initial conditions for the N-body run from the Schwarzschild model? Zhao: I select the particles so that the number of particles on each orbit is proportional to the weight on that orbit derived from the Schwarzschild method. Inside each orbit, particles are randomly distributed in phase. So the N-body run is based on a random realization of the bar orbits.

A self-consistent stellar dynamical model for the Galactic bar is constructed from about 500 numerically computed orbits with an extension of the Schwarzschild technique. The model fits the {\sl COBE} found asymmetric boxy light distribution and the observ

Ben Werner: The match of the x1 orbits to the tangent point pro le in the HI v diagram is impressive (and good because it does away with the dip in the Clemens et al. rotation curve), but so far it doesn't get enough material into the forbidden regions (I think that is because the bar is so close to end-on in your model.) Zhao: It is true that the x1 orbits in the best model do not fully occupy the forbidden region. But I suspect that is because full hydrodynamics of the gas has not been included in the potential model.l

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