Mem. S.A.It. Vol. 00, 189 c SAIt 2008 Memorie della New method for determining the Milky Way bar pattern speed I. Minchev1, J. Nordhaus2 and A. C. Quillen3 1 Observatoire Astronomique, Universit´ede Strasbourg, Strasbourg, France 2 Department of Astrophysical Sciences, Princeton University, Princeton, NJ, USA 3 Department of Physics and Astronomy, University of Rochester, Rochester, NY, USA Abstract. Previous work has related the Galactic bar to structure in the local stellar ve- locity distribution. Here we show that the bar also influences the spatial gradients of the velocity vector via the Oort constants. By numerical integration of test-particles we sim- ulate measurements of the Oort C value in a gravitational potential including the Galactic bar. We account for the observed trend that C is increasingly negative for stars with higher velocity dispersion. By comparing measurements of C with our simulations we improve on previous models of the bar, estimating that the bar pattern speed is Ωb/Ω0 = 1.87 ± 0.04, ◦ ◦ where Ω0 is the local circular frequency, and the bar angle lies within 20 6 φ0 6 45 . We find that the Galactic bar affects measurements of the Oort constants A and B less than ∼ 2 km s−1 kpc−1 for the hot stars. 1. Introduction as orientation and pattern speed, have only been inferred indirectly from observations of The Galaxy is often modeled as an ax- the inner Galaxy (e.g., Blitz & Spergel 1991; isymmetric disk. With the ever increasing Weinberg 1992). proper motion and radial velocity data, it is now apparent that non-axisymmetric effects However, the bar has also been found to cannot be neglected (nonzero Oort constant affect the local velocity distribution of stars. C, Olling & Dehnen 2003, hereafter OD03; HIPPARCOS data revealed more clearly a nonzero vertex deviation, Famaey et al. 2005; stream of old disk stars with an asymmetric asymmetries in the local velocity distribution drift of about 45 km s−1 and a radial velocity of stars, Dehnen 1998). It is still not clear, u < 0, with u and v positive toward the Galactic however, what the exact nature of the perturb- center and in the direction of Galactic rota- ing agent(s) in the Solar neighborhood (SN) tion, respectively. This agglomeration of stars is(are). Possible candidates are spiral density has been dubbed the ‘Hercules’ stream or the waves, a central bar, and a triaxial halo. ‘u-anomaly’. The numerical work of Dehnen (1999, 2000) and Fux (2001) has shown that arXiv:1002.1742v1 [astro-ph.GA] 9 Feb 2010 Due to our position in the Galactic disk, the properties of the Milky Way bar are hard this stream can be explained as the effect of the to observe directly. Hence its parameters, such Milky Way bar if the Sun is placed just out- side the 2:1 Outer Lindblad Resonance (OLR). Send offprint requests to: Ivan Minchev; e-mail: Using high-resolution spectra of nearby F and [email protected] G dwarf stars, Bensby et al. (2007) have inves- 190 Minchev, Nordhaus, and Quillen: Milky Way bar pattern speed tigated the detailed abundance and age struc- where d is the average heliocentric distance of ture of the Hercules stream. Since this stream is stars, A and B are the usual Oort constants, and composed of stars of different ages and metal- C and K are given by licities pertinent to both thin and thick disks, u ∂u 1 ∂vφ they concluded that a dynamical effect, such as 2C ≡ − + − (2) the influence of the bar, is a more likely expla- r ∂r r ∂φ nation than a dispersed cluster. u ∂u 1 ∂vφ 2K ≡ + + + . (3) Assuming the Galactic bar affects the shape r ∂r r ∂φ of the distribution function of the old stellar population in the SN, an additional constraint Here r and φ are the usual polar coordinates on the bar can be provided by considering the and vφ = v0 + v, where v0 is the circular veloc- values derived for the Oort constant C. In other ity at the Solar radius, r0. In this work we pri- words, in addition to relating the dynamical in- marily consider a flat rotation curve (RC; see, fluence of the bar to the local velocity field, C however §5), hence the derivatives of vφ in the provides a link to the gradients of the velocities above equations are identical to the derivatives as well. The study of OD03 not only measured of v. C describes the radial shear of the velocity a non-zero C, implying the presence of non- field and K its divergence. For an axisymmet- circular motion in the SN, but also found that C ric Galaxy we expect vanishing values for both 1 is more negative for older and redder stars with C and K . Whereas C could be derived from a larger velocity dispersion. This is surpris- both radial velocities and proper motions, K ing as a hotter stellar population is expected to can only be measured from radial velocities, in have averaged properties more nearly axisym- which case accurate distances are also needed. metric and hence, a reduced value of |C| (e.g., A problem with using proper motions data Minchev & Quillen 2007; hereafter Paper I). has been described by OD03. The authors It is the aim of this work to show it is present an effect which arises from the longi- indeed possible to explain the observationally tudinal variations of the mean stellar parallax deduced trend for C (OD03) by modeling the caused by intrinsic density inhomogeneities. Milky Way as an exponential disk perturbed Together with the reflex of the solar motion by a central bar. By performing this exercise, these variations create contributions to the lon- we provide additional constraints on the bar’s gitudinal proper motions which are indistin- pattern speed. Models for the structure in the guishable from the Oort constants at ≤ 20% bulge of our galaxy are difficult to constrain of their amplitude. OD03 corrected for the because of the large numbers of degrees of ‘mode-mixing’ effect described above, using freedom in bar models. Thus future studies of the latitudinal proper motions. The resulting C structure in the Galactic Center will benefit is found to vary approximately linearly with from tighter constraints on the parameters de- both color and asymmetricdrift (and thus mean age) from C ≈ 0 km s−1 kpc−1 for blue sam- scribing the bar, such as its pattern speed and −1 −1 angle with respect to the Sun. ples to C ≈ −10 km s kpc for late-type stars (see Figs. 6 and 9 in OD03).Since C is re- lated to the radial streaming of stars, we expect ff 2. The Oort constants non-axisymmetric structure to mainly a ect the low-dispersion population which would re- We can linearize the local velocity field (e.g., sult in the opposite behavior for C. In Paper Paper I) about the LSR and write the mean ra- I we showed that spiral structure failed to ex- dial velocity vd and longitudinal proper motion plain the observed trend of C. µl as functions of the Galactic longitude l as Note that the Oort constants are not con- stant unless they are measured in the SN. Due vd = K + A sin(2l) + C cos(2l) (1) 1 Note, however, that C and K would also be zero d in the presence of non-axisymmetric structure if the µl = B + A cos(2l) − C sin(2l) Sun happened to be located on a symmetry axis. Minchev, Nordhaus, and Quillen: Milky Way bar pattern speed 191 Table 1. Simulation parameters used Parameter Symbol Value Solar neighborhood radius r0 1 Circular velocity at r0 v0 1 Radial velocity dispersion σu(r0) 0.05v0 or 0.18v0 σu scale length rσ 0.9r0 Disk scale length rρ 0.37r0 Bar strength ǫb −0.012 Bar size rb 0.8rcr to non-axisymmetries they may vary with the rameter used by Dehnen (2000); the strength position in the Galaxy. Thus, the Oort con- is specified by ǫb = −α from the same pa- stants have often been called the Oort func- per. The bar length is rb = 0.8rcr with rcr tions. the bar corotation radius. The pattern speed, Ωb is kept constant. We grow the bar by lin- 3. The simulations early varying its amplitude, ǫb, from zero to its maximum value in four bar rotation peri- We perform 2D test-particle simulations of ods. We present our results by changing Ωb and an initially axisymmetric exponential galactic keeping r0 fixed. The 2:1 outer Lindblad res- disk. To reproduce the observed kinematics of onance (OLR) with the bar is achieved when the Milky Way, we use disk parameters con- Ωb/Ω0 = 1+κ/2 ≈ 1.7, where κ is the epicyclic sistent with observations (see Table 1). A de- frequency. We examine a region of parame- tailed description of our disk model and sim- ter space for a range of pattern speeds placing ulation technique can be found in Paper I. We the SN just outside the OLR. For a given pat- are interested in the variation of C with color tern speed one could obtain the ratio r0/rOLR (B − V), and asymmetric drift va. We simulate through r0/rOLR =Ωb/ΩOLR ≈ Ωb/1.7. variations with color by assuming that the ve- In contrast to Dehnen (2000) and simi- locity dispersion increases from blue to red star lar to Fux (2001) and M¨uhlbauer & Dehnen samples. For a cold disk we start with an initial (2003), we integrate forward in time.