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Register a new userManaging resonant trapped orbits in our Galaxy
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James Binney
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Galaxy modelling is greatly simplified by assuming the existence of a global system of angle-action coordinates. Unfortunately, global angle-action coordinates do not exist because some orbits become trapped by resonances, especially where the radial and vertical frequencies coincide. We show that in a realistic Galactic potential such trapping occurs only on thick-disc and halo orbits (speed relative to the guiding centre >~80 km/s). We explain how the Torus Mapper code (TM) behaves in regions of phase space in which orbits are resonantly trapped, and we extend TM so trapped orbits can be manipulated as easily as untrapped ones. The impact that the resonance has on the structure of velocity space depends on the weights assigned to trapped orbits. The impact is everywhere small if each trapped orbit is assigned the phase space density equal to the time average along the orbit of the DF for untrapped orbits. The impact could be significant with a different assignment of weights to trapped orbits.
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Torus mapping yields constants of motion for stars trapped at a resonance. Each such constant of motion yields a system of contours in velocity space at the Sun and neighbouring points. If Jeans theorem applied to resonantly trapped orbits, the density of stars in velocity space would be equal at all intersections of any two contours. A quantitative measure of the violation of this principal is defined and used to assess various pattern speeds for a model of the bar recently fitted to observations of interstellar gas. Trapping at corotation of a bar with pattern speed in the range 33-36 /Gyr is favoured and trapping at the outer Lindblad resonance is disfavoured. As one moves around the Sun the structure of velocity space varies quite rapidly, both as regards the observed star density and the zones of trapped orbits. The data seem consistent with trapping at corotation.
Accelerations of both the solar system barycenter (SSB) and stars in the Milky Way cause a systematic observational effect on the stellar proper motions, which was first studied in the early 1990s and developed by J. Kovalevsky (aberration in proper motions, 2003, A&A, 404, 743). This paper intends to extend that work and aims to estimate the magnitude and significance of the aberration in proper motions of stars, especially in the region near the Galactic center. We adopt two models for the Galactic rotation curve to evaluate the aberrational effect on the Galactic plane. Based on the theoretical developments, we show that the effect of aberration in proper motions depends on the galactocentric distance of stars; it is dominated by the acceleration of stars in the central region of the Galaxy. Within 200 pc from the Galactic center, the systematic proper motion can reach an amplitude larger than 1000 uas/yr by applying a flat rotation curve. With a more realistic rotation curve which is linearly rising in the core region of the Galaxy, the aberrational proper motions are limited up to about 150 uas/yr. Then we investigate the applicability of the theoretical expressions concerning the aberrational proper motions, especially for those stars with short period orbits. If the orbital period of stars is only a fraction of the light time from the star to the SSB, the expression proposed by Kovalevsky is not appropriate. With a more suitable formulation, we found that the aberration has no effect on the determination of the stellar orbits on the celestial sphere. The aberrational effect under consideration is small but not negligible with high-accurate astrometry in the future, particularly in constructing the Gaia celestial reference system realized by Galactic stars.
The conventional approach to orbit trapping at Lindblad resonances via a pendulum equation fails when the parent of the trapped orbits is too circular. The problem is explained and resolved in the context of the Torus Mapper and a realistic Galaxy model. Tori are computed for orbits trapped at both the inner and outer Lindblad resonances of our Galaxy. At the outer Lindblad resonance, orbits are quasiperiodic and can be accurately fitted by torus mapping. At the inner Lindblad resonance, orbits are significantly chaotic although far from ergodic, and each orbit explores a small range of tori obtained by torus mapping.
The spiral structure of our Milky Way Galaxy is not yet known. HII regions and giant molecular clouds are the most prominent spiral tracers. We collected the spiral tracer data of our Milky Way from the literature, namely, HII regions and giant molecular clouds (GMCs). With weighting factors based on the excitation parameters of HII regions or the masses of GMCs, we fitted the distribution of these tracers with models of two, three, four spiral-arms or polynomial spiral arms. The distances of tracers, if not available from stellar or direct measurements, were estimated kinetically from the standard rotation curve of Brand & Blitz (1993) with $R_0$=8.5 kpc, and $Theta_0$=220 km s$^{-1}$ or the newly fitted rotation curves with $R_0$=8.0 kpc and $Theta_0$=220 km s$^{-1}$ or $R_0$=8.4 kpc and $Theta_0$=254 km s$^{-1}$. We found that the two-arm logarithmic model cannot fit the data in many regions. The three- and the four-arm logarithmic models are able to connect most tracers. However, at least two observed tangential directions cannot be matched by the three- or four-arm model. We composed a polynomial spiral arm model, which can not only fit the tracer distribution but also match observed tangential directions. Using new rotation curves with $R_0$=8.0 kpc and $Theta_0$=220 km s$^{-1}$ and $R_0$=8.4 kpc and $Theta_0$=254 km s$^{-1}$ for the estimation of kinematic distances, we found that the distribution of HII regions and GMCs can fit the models well, although the results do not change significantly compared to the parameters with the standard $R_0$ and $Theta_0$.
The nature of our Milky Way Galaxy is reexamined from an eclectic point of view. Evidence for a central bar, for example, is not reflected in the distribution of RR Lyrae variables in the central bulge [4,5], and it is not clear if either a 2-armed or 4-armed spiral pattern is appropriate for the spiral arms. Radial velocity mapping of the Galaxy using radio H I, H II, or CO observations is compromised by the assumptions adopted for simple Galactic rotation. The Suns local standard of rest (LSR) velocity is $sim 14$ km s$^{-1}$ rather than 20 km s$^{-1}$, the local circular velocity is $251 pm 9$ km s$^{-1}$ rather than 220 km s$^{-1}$, and young groups of stars exhibit a 10--20 km s$^{-1}$ kick relative to what is expected from Galactic rotation. By implication, the same may be true for star-forming gas clouds affected by the Galaxys spiral density wave, raising concerns about their use for mapping spiral arms. Proper motion data in conjunction with the newly-estimated velocity components for the Suns motion imply a distance to the Galactic centre of $R_0=8.34pm0.27$ kpc, consistent with recent estimates which average $8.24pm0.09$ kpc. A cosinusoidal Galactic potential is not ruled out by observations of open star clusters. The planetary nebula cluster Bica 6, for example, has a near-escape orbit for a Newtonian potential, but a near-normal orbit in a cosinusoidal potential field. The nearby cluster Collinder 464 also displays unusually large tidal effects consistent with those expected for a cosinusoidal potential. A standard Newtonian version of the Virial Theorem for star clusters yields very reasonable masses ($sim 3 times 10^{11}M_{odot}$ and $sim 4 times 10^{11}M_{odot}$) for the Milky Way and M31 subgroups of the Local Group, respectively. A cosinusoidal relation should yield identical results.
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