arXiv:astro-ph/0703805v3 11 Dec 2007 ohse, 42;[email protected] 14623; Rochester, ohse,Rcetr Y167 [email protected]. 14627; NY [email protected] Rochester, Rochester, aia ffc,sc steiflec fteBr samore a cluster. dy- is dispersed , a a the that than of concluded influence explanation they the likely com- as disks, such is thick effect, and stream namical pertinent metallicities thin this and both ages Since different to of stream. (2007) stars of al. Hercules struc- posed et the age Bensby and of stars, abundance ture dwarf detailed Outer the and 2:1 investigated have the Milky nearby outside spec- the of high-resolution just of tra Using placed effect (OLR). is Resonance the Sun that Lindblad as the shown if explained has Bar be (2001) Way Fux can and stream 2000) this (1999, Dehnen of hsagoeaino tr a endbe “Her- the “ dubbed the been or has stream stars respectively.cules” rotation, of Galactic agglomeration of direction This velocity the radial in an and a ter with and stars km/ < disk 45 u old about of of drift stream asymmetric a clearly more revealed oa eoiydsrbto fstars. of distribution velocity Weinberglocal 1991; Spergel & of Blitz observations (.g., speed, 1992). from Galaxy pattern indirectly inner inferred and the orientation been only Hence as directly. have such observe to parameters, hard are its Bar Way Milky the of a and Bar, central a halo. waves, triaxial density Possible perturbing spiral is(are). the are (SN) candidates of neighborhood Solar nature the exact not in still agent(s) the is local what It the however, 1998). in clear, Dehnen vertex stars, asymmetries of nonzero 2005; distribution velocity &; al. constant et hereafter nonaxisymmetric Famaey Oort 2003, deviation, that (nonzero Dehnen apparent & neglected ve- Olling now radial be is and cannot motion it effects proper data, increasing locity ever the With 2 rpittpstuigL using 2018 typeset 7, Preprint November version Draft 1 oee,teBrhsas enfudt ffc the affect to found been also has Bar the However, properties the disk, Galactic the in position our to Due disk. axisymmetric an as modeled often is Galaxy The aoaoyfrLsrEegtc,Uiest fRochester, of University Energetics, Laser for Laboratory eateto hsc n srnm,Uiest of University Astronomy, and Physics of Department ,with 0, eew hwta h a loiflecstesailgainso of gradients simulate spatial we the test-particles influences of also integration Bar numerical the By that constants. show we Here sicesnl eaiefrsaswt ihrvlct iprin yc By fo dispersion. account velocity We higher with Bar. stars Galactic for the negative including increasingly potential is gravitational a in 20 ihorsmltosw mrv npeiu oeso h a,estimat Bar, the of models previous on Ω improve is we speed simulations our with ujc headings: Subject esthan less rvoswr a eae h aatcBrt tutr ntelocal the in structure to Bar Galactic the related has work Previous ◦ 6 u φ and 0 6 ∼ 1. / 45 mskcfrtehtstars. hot the for km/s/kpc 2 Ω A INTRODUCTION T oiietwr h aatccen- Galactic the toward positive u ◦ 0 E efidta h aatcBraet esrmnso h otco Oort the of measurements affects Bar Galactic the that find We . aoay.Tenmrclwork numerical The -anomaly”. tl mltajv 08/22/09 v. emulateapj style 1 = . 87 ± E OSRIT NTEGLCI BAR GALACTIC THE ON CONSTRAINTS NEW .Minchev I. 0 . 4 hr Ω where 04, HIPPARCOS 1 rf eso oebr7 2018 7, November version Draft .Nordhaus . , 0 stelclcrua rqec,adteBragele within lies angle Bar the and frequency, circular local the is data ABSTRACT edu, , 1,2 where ie by given n ogtdnlpoe motion proper longitudinal and and aatclongitude Galactic )aotteLRadwietema ailvelocity radial mean the write and LSR the about I) respect with angle and Sun. speed the pattern its describing to as parameters such the Bar, fu- bene- on the will Thus constraints Center tighter Galactic models. from the Bar fit in large structure in the of of freedom of studies bulge of ture because the degrees constrain in of to this structure numbers difficult performing the are Bar’s galaxy for By the our Models on constraints speed. Bar. exponential additional pattern central an provide as a we Way by exercise, for Milky perturbed trend the disk deduced modeling observationally by the (O&D) explain to ble of value reduced a hence, | and hotter axisymmetric a properties averaged nearly as have more to surprising expected is is population This stellar dispersion. velocity larger rsneo o-iclrmto nteS,btas found also but SN, the non-zero that in a study motion The non-circular measured of well. only presence as not velocities the O&D of of gradients the to link a nuneo h a otelclvlct field, velocity local the to constant Bar the Oort of the influence for provided derived be values can C Bar the the the considering in on by population constraint stellar additional an old SN, the of function distribution C n .C Quillen C. A. and nohrwrs nadto orltn h dynamical the relating to addition in words, other In . ecnlnaietelclvlct ed(.. Paper (e.g., field velocity local the linearize can We possi- indeed is it show to letter this of aim the is It suigteGlci a ffcstesaeo the of shape the affects Bar Galactic the Assuming | eg,Mnhv&Qiln20;hratrPprI). Paper hereafter 2007; Quillen & Minchev (e.g., B C d smr eaiefrodradrde tr iha with stars redder and older for negative more is h sa otcntns and constants, Oort usual the are steaeaehloeti itneo stars, of distance heliocentric average the is esrmnso h Oort the of measurements v µ d d 2. 2 = = evlct etrvateOort the via vector velocity he C H OTCONSTANTS OORT THE maigmaueet of measurements omparing B l h bevdtedthat trend observed the r − ≡ tla eoiydistribution. velocity stellar 1 + as + n htteBrpattern Bar the that ing A A u r cos(2 sin(2 + ∂ ∂r u l l ) + ) nstants − − µ 1 r C C l ∂ ∂φ sin(2 cos(2 sfntoso the of functions as v φ A C C l and l mligthe implying , ) (1) ) value C and C B C C provides K are (2) v C A d 2

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 u ∂u 1 ∂v 2K ≡ + + + φ . (3) r ∂r r ∂φ

Here r and φ are the usual polar coordinates and vφ = v0+v, where v0 is the circular velocity at the Solar radius, r0. In this work we primarily consider a flat rotation curve (RC) (see, however §5), hence the derivatives of vφ in the above equations are identical to the derivatives Fig. 1.— Simulated u − v distribution for a Bar pattern speed ◦ of v. C describes the radial shear of the velocity field Ωb = 1.87Ω0, Bar angle φ0 = 35 and a sample depth d = r0/40. and K its divergence. For an axisymmetric Galaxy we The initial velocity dispersion is σu = 0.18v0 and the Bar strength expect vanishing values for both C and K 3. Whereas C is ǫb = −0.012. Contour levels are equally spaced. The clump iden- tified with the Hercules stream is clearly discernible in the lower could be derived from both radial velocities and proper left portion of the plot. This figure is in agreement with the ob- motions, K can only be measured from radial velocities, served f0(u, v) (Dehnen 1998; Fux 2001) and simulated distribution in which case accurate distances are also needed. functions of Dehnen (2000) and Fux (2001). A problem with using proper motions data has been disk we use σu =0.18v0. The background axisymmetric described by O&D. The authors present an effect which potential due to the disk and halo has the form Φ0(r)= arises from the longitudinal variations of the mean stel- 2 v0 log(r), corresponding to a flat RC. lar parallax caused by intrinsic density inhomogeneities. Together with the reflex of the solar motion these varia- 3.1. The Bar potential tions create contributions to the longitudinal proper mo- We model the nonaxisymmetric potential perturbation tions which are indistinguishable from the Oort constants due to the Galactic Bar as a pure quadrupole at ≤ 20% of their amplitude. O&D corrected for the “mode-mixing” effect described above, using the latitu- rb 3  r , r ≥ rb dinal proper motions. The resulting C is found to vary Φb = Ab(ǫb) cos[2(φ − Ωbt)] ×  3 approximately linearly with both color and asymmetric  2 − r , r ≤ r  rb  b drift (and thus mean age) from C ≈ 0 km/s/kpc for  (4) blue samples to C ≈ −10 km/s/kpc for late-type stars Here A (ǫ ) is the Bar’s gravitational potential ampli- (see Figs. 6 and 9 in O&D). Since C is related to the b b tude, identical to the same name parameter used by radial streaming of stars, we expect non-axisymmetric Dehnen (2000); the strength is specified by ǫ = −α structure to mainly affect the low-dispersion population b from the same paper. The Bar length is r = 0.8r which would result in the opposite behavior for C. In b cr with r the Bar corotation radius. The pattern speed, Paper I we showed that spiral structure failed to explain cr Ω is kept constant. We grow the Bar by linearly vary- the observed trend of C. b ing its amplitude, ǫ , from zero to its maximum value Note that the Oort constants are not constant unless b in four Bar rotation periods. We present our results by they are measured in the SN. Due to nonaxisymmetries changing Ω and keeping r fixed. The 2:1 outer Lind- they may vary with the position in the Galaxy. Thus, the b 0 blad resonance (OLR) with the Bar is achieved when Oort constants have often been called the Oort functions. Ωb/Ω0 = 1+ κ/2 ≈ 1.7, where κ is the epicyclic fre- 3. THE SIMULATIONS quency. We examine a region of parameter space for a range of pattern speeds placing the SN just outside the We perform 2D test-particle simulations of an initially OLR. For a given pattern speed one could obtain the axisymmetric exponential galactic disk. To reproduce ratio r0/rOLR through r0/rOLR =Ωb/ΩOLR ≈ Ωb/1.7. the observed kinematics of the Milky Way, we use disk In contrast to Dehnen (2000) and similar to Fux (2001) parameters consistent with observations (see Table 1). and ¨uhlbauer & Dehnen (2003), we integrate forward A detailed description of our disk model and simulation in time. To allow for phase mixing to complete after technique can be found in Paper I. We are interested in the Bar reaches its maximum strength, we wait for 10 the variation of C with color (B−V ), and asymmetric Bar rotations before we start recording the position and drift va. We simulate variations with color by assuming velocity vectors. In order to improve statistics, positions that the velocity dispersion increases from blue to red and velocities are time averaged for 10 Bar periods. After star samples. For a cold disk we start with an initial utilizing the two-fold symmetry of our galaxy we end radial velocity dispersion σu = 0.05v0 whereas for a hot up with ∼ 5 × 105 particles in a given simulated solar 3 neighborhood with maximum radius d = r /10. Note, however, that C and K would also be zero in the presence max 0 of nonaxisymmetric structure if the Sun happened to be located on 4. RESULTS a symmetry axis. 3

Fig. 2.— Each panel shows the variation of the Oort constant C with Bar angle φ0, for a simulation with the parameters given in Table 1 and a particular Bar pattern speed, Ωb, and a mean sample depth, d. Solid and dotted lines correspond to cold- and hot-disk values, respectively. Columns from left to right, show an increasing Ωb in units of Ω0. Note that the OLR is at ΩOLR ≈ 1.7. Different rows present results from samples with different mean heliocentric distance, d. Good matches to the observed trend in C (vanishing value for the cold ◦ ◦ disk and a large negative for the hot one) are achieved for 20 6 φ0 6 45 and 1.83 6 Ωb/Ω0 6 1.91.

First we show that we can reproduce the results of write and Cc to refer to the values for C as estimated Dehnen and Fux for the Hercules stream. We present a from the hot and cold disks, respectively. simulated u − v velocity distribution for Ωb = 1.87Ω0, Ch (dotted lines in Fig. 3.1) varies with galactic az- ◦ φ0 = 35 , and a sample depth d = r0/40 in fig. 1. The imuth as Ch(φ0) ∼ sin(2φ0) for all of the Ωb values con- initial velocity dispersion is σu =0.18v0. This plot is in- sidered. On the other hand, the cold disk values (solid deed in very good agreement with previous test-particle line) exhibit different variations, depending on Ωb or (Dehnen 2000; Fux 2001) and N-body (Fux 2001) simu- equivalently, on the ratio r0/rOLR. Closer to the OLR lations. (left columns of Fig. 3.1), Cc(φ0) approaches the func- tional behavior of Ch(φ0). Away from the OLR Cc(φ0) By a quantitative comparison of the observed with ◦ the simulated distributions, Dehnen (2000) deduced the is shifted by 90 compared to Ch(φ0). While both cold Milky Way Bar pattern speed to be Ωb/Ω0 =1.85±0.15. and hot disks yield in increase in the amplitude of C(φ0) In this calculation only the local velocity field, i.e., the as the pattern speed nears the OLR, the effect on the observed u−v distribution, was taken into account. Now, Cc(φ0) is much stronger. This is consistent with our if one could also relate the dynamical effect of the Bar to expectation that the cold disk is affected more by the the derivatives of the velocities, an additional constraint Bar, especially near the OLR. While close to the OLR on Bar parameters would be provided. Since the gradi- |Ch(φ0)| < |Cc(φ0)|, we observe the opposite behavior ents of u and v are hard to measure directly, one needs away from it. This could be explained by the results of an indirect way of achieving this task. The obvious can- M¨uhlbauer & Dehnen (2003), where they find that high didates, as anticipated, are the Oort constants C and velocity dispersion stars tend to shift the “effective reso- K. Oort’s K, however, is hard to measure as mentioned nance” radially outwards. above; hence only C, as estimated by O&D, will be em- According to O&D’s “mode-mixing” corrected value, ployed here. By Fourier expansion of eqs. 1 (see Paper C ≈ 0 for the cold sample and decreases to about -10 I for details) we estimate C from our numerically simu- km/s for the hot population. Hence we look for loca- lated velocity distributions for the initially cold and hot tions in Fig. 3.1 complying with this requirement. In disks. the transition region, 1.83 6 Ωb/Ω0 6 1.91, where the function C (φ ) transforms into its negative, there are 4.1. c 0 Variation of C with Bar pattern speed and specific angles which provide good matches for the ob- orientation servations, i.e., Cc(φ0) ≈ 0 while Ch(φ0) is significantly In Fig. 3.1 we present our results for C as a function negative. We performed simulations with different pat- of the Bar angle, φ0 (the angle by which the Sun’s az- tern speeds in the range 1.7 6 Ωb/Ω0 6 2.5. It was imuth lags the Bar’s major axis). Each column shows found that only in the transition region in Fig. 3.1 could a simulation with a different pattern speed, indicated in one achieve a satisfactory match to the observations. Al- each plot. Rows from top to bottom show C as calcu- though the left- and right-most columns are nearly con- lated from samples at average heliocentric distances cor- sistent with our requirement on C, we reject these values responding to d = 200, d = 400, and d = 600 pc, for a of Ωb for the following reason. O&D estimated a large Solar radius r0 = 7.8 kpc. Solid and dotted lines repre- negative C from both the reddest main-sequence stars, sent the results for cold and hot disks, respectively. The which are nearby, and the more distant red giants. This dashed lines indicates C = 0. C is presented in units of is inconsistent with the largest mean sample depth shown Ω0 = v0/r0. To make the discussion less cumbersome, we in Fig. 3.1 with Ωb = 1.81, 1.93. We conclude that the 4

Bar pattern speed must lie in the range ing Ωb/Ω0 = 1.87 ± 0.04. In addition, the Bar angle is ◦ ◦ found to be in the range 20 6 φ0 6 45 . Our result Ωb/Ω0 =1.87 ± 0.04. (5) for Ωb lies well within the estimate by Dehnen (2000) The Bar angle with respect to the Sun has been pro- and that by Debattista et al. (2002), based on OH/IR posed, as derived from IR photometry, to lie in the range star kinematics. This study provides an improvement on 15◦ − 45◦ with the Sun lagging the Bar major axis, the measurement of the Bar pattern speed by a factor of whereas Dehnen (2000) found that the Bar reproduced ∼ 4 compared to previous work (Dehnen 2000). The im- ◦ ◦ the u-anomaly for 10 6 φ0 6 70 . Examining Fig. 3.1, proved constraints on Bar parameters should be tested we find that our requirements on C result in constraining by, and will benefit, future studies of the Bar structure ◦ ◦ the Bar angle in the range 20 6 φ0 6 45 , depending in the Galactic center region, that will become possi- on Ωb. This result for the dependence of pattern speed ble with future radial velocity and proper motion studies on Bar angle is in a very good agreement with Fig. 10 (e.g., BRAVA, Rich et al. 2007). In addition to a flat RC we have also considered a by Dehnen (2000) which shows a linear increase of the β derived Ωb/Ω0 as a function of φ0. This is remarkable power law initial tangential velocity vφ = v0(r/r0) with since Dehnen (2000) obtained his results in a completely β = 0.1, −0.1 corresponding to a rising and a declining different way than our method here. RC, respectively. We found that for β = 0.1 no addi- tional error in Ωb is introduced. However, for β = −0.1 4.2. The effect of the Bar on A and B we estimated Ωb/Ω0 =1.85 ± 0.06. From Fig. 3.1 we see that the main source of error in the Bar pattern speed is The Oort constants A and B were found to also be due to the uncertainty in the Bar angle. A recent work affected by the Bar, although to a much smaller ex- by Rattenbury et al. (2007) used OGLE-II microlensing tent. One could estimate the Bar-induced errors as observations of red clump giants in the Galactic bulge, ∆A ≡ A − Aaxi and ∆B ≡ B − Baxi. Here Aaxi, Baxi are to estimate a bar angle of 24◦ − 27◦. Using this as an the resulting axisymmetric values when simulations ex- additional constraint we obtain Ωb/Ω0 =1.84 ± 0.01. cluding the Bar perturbation are performed. Note that While we account for the trend of increasingly nega- we require a different simulation with each of the initial tive C with increasing velocity dispersion, we fail to re- radial velocity dispersions used in order to estimate the produce the size of the C-value measured by O&D. The asymmetric drift induced errors. We found that for the most negative value we predict is about -6 while they cold disk the Bar causes ∆A =3.5±0.7 and ∆B =3±0.6 measure -10 km/s/kpc. We discuss possible reasons for km/s/kpc, while the hot disk yielded ∆A =0.5±1.5 and this discrepancy: (i) we expect that the magnitude of ∆B = 0 ± 1 km/s/kpc. These values are calculated for C is related to the fraction of SN stars composing the r0 =7.8 kpc and v0 = 220 km/s. Despite the Bar’s effect Hercules stream. It is possible that during its formation on C, the hot population does provide better measure- the Bar changed its pattern speed forcing more stars to ments for A and B. be trapped in the 2:1 OLR, thus increasing this frac- O&D found an approximately linear variation of A and tion. This would give rise to a more negative C from hot B with color with A increasing and B decreasing. The stars, while leaving the cold population unaffected; (ii) Bar induced errors presented here match the observed the numerical model here uses a distance limited sample. trend of B but have the opposite one for A. It is possible Modeling a stellar population with a magnitude limited that local spiral structure is responsible for this discrep- sample is more appropriate to compare to the observed ancy. measurements. (iii) O&D’s “mode-mixing” correction is 5. CONCLUSION only valid if the Sun’s motion in the -direction were as- sumed constant. This is invalidated if, for example, a We have shown that the Galactic Bar can account for local Galactic warp were present. a trend seen in measurements of Oort’s C, namely a more negative C-value with increasing velocity disper- sion. This trend is difficult to explain with other dy- Support for this work was in part provided namical models (e.g., Paper I). By requiring that the by NSF grant ASST-0406823, and NASA grant Bar model accounts for this trend in C, we have im- No. NNG04GM12G issued through the Origins of Solar proved the measurement of the Bar pattern speed, find- Systems Program.

REFERENCES Bensby, T., Oey, M. S., Feltzing, S., & Gustafsson, B. 2007, ApJ, Minchev, I., & Quillen, A. C. 2006, MNRAS, 368, 623 655, L89 Minchev, I., & Quillen, A. C. 2007, MNRAS, 377, 1163 (Paper I) Blitz, L., & Spergel, D. N. 1991, ApJ, 379, 631 M¨uhlbauer, G., & Dehnen, . 2003, A&A, 401, 975 Debattista, V. P., Gerhard, O., & Sevenster, M. N. 2002, Olling, R. P., & Dehnen, W. 2003, ApJ, 599, 275 (O&D) MNRAS, 334, 355 Rattenbury, N. J., Mao, S., Sumi, T., & Smith, M. C. 2007, Dehnen, W. 1998, AJ, 115, 2384 ArXiv e-prints, 704, arXiv:0704.1614 Dehnen, W. 1999, ApJ, 524, L35 Rich, R. M., Reitzel, D. B., Howard, C. D., & Zhao, H. 2007, Dehnen, W. 2000, AJ, 119, 800 ApJ, 658, L29 Famaey, B., Jorissen, A., Luri, X., Mayor, M., Udry, S., Torra, J., Fern´andez, D., & Figueras, F. 2000, A&A, 359, 82 Dejonghe, H., & Turon, C. 2005, A&A, 430, 165 Weinberg, M. D. 1992, ApJ, 384, 81 Fux, R. 2001, A&A, 373, 511 Kuijken, K., & Tremaine, S. 1991, in Sundelius B., ed., Dynamics of Disc Galaxies, Gteborg Univ. Press, p. 71 Draft version December 11, 2007 A Preprint typeset using LTEX style emulateapj v. 11/26/04

NEW CONSTRAINTS ON THE GALACTIC BAR I. Minchev1, J. Nordhaus1,2 and A. C. Quillen1 Draft version December 11, 2007

ABSTRACT Previous work has related the Galactic Bar to structure in the local stellar velocity 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 simulate 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/kpc for the hot stars. Subject headings:

1. INTRODUCTION Assuming the Galactic Bar affects the shape of the The Galaxy is often modeled as an axisymmetric disk. distribution function of the old stellar population in the With the ever increasing proper motion and radial ve- SN, an additional constraint on the Bar can be provided locity data, it is now apparent that nonaxisymmetric by considering the values derived for the Oort constant effects cannot be neglected (nonzero Oort constant C, C. In other words, in addition to relating the dynamical Olling & Dehnen 2003, hereafter O&D; nonzero vertex influence of the Bar to the local velocity field, C provides deviation, Famaey et al. 2005; asymmetries in the local a link to the gradients of the velocities as well. The study velocity distribution of stars, Dehnen 1998). It is still not of O&D not only measured a non-zero C, implying the clear, however, what the exact nature of the perturbing presence of non-circular motion in the SN, but also found agent(s) in the Solar neighborhood (SN) is(are). Possible that C is more negative for older and redder stars with a candidates are spiral density waves, a central Bar, and a larger velocity dispersion. This is surprising as a hotter triaxial halo. stellar population is expected to have averaged properties Due to our position in the Galactic disk, the properties more nearly axisymmetric and hence, a reduced value of of the Milky Way Bar are hard to observe directly. Hence |C| (e.g., Minchev & Quillen 2007; hereafter Paper I). its parameters, such as orientation and pattern speed, It is the aim of this letter to show it is indeed possi- have only been inferred indirectly from observations of ble to explain the observationally deduced trend for C the inner Galaxy (e.g., Blitz & Spergel 1991; Weinberg (O&D) by modeling the Milky Way as an exponential 1992). disk perturbed by a central Bar. By performing this However, the Bar has also been found to affect the exercise, we provide additional constraints on the Bar’s local velocity distribution of stars. HIPPARCOS data pattern speed. Models for the structure in the bulge of revealed more clearly a stream of old disk stars with an our galaxy are difficult to constrain because of the large asymmetric drift of about 45 km/s and a radial velocity numbers of degrees of freedom in Bar models. Thus fu- u < 0, with u and v positive toward the Galactic cen- ture studies of structure in the Galactic Center will bene- ter and in the direction of Galactic rotation, respectively. fit from tighter constraints on the parameters describing This agglomeration of stars has been dubbed the “Her- the Bar, such as its pattern speed and angle with respect cules” stream or the “u-anomaly”. The numerical work to the Sun. of Dehnen (1999, 2000) and Fux (2001) has shown that 2. THE OORT CONSTANTS this stream can be explained as the effect of the Milky We can linearize the local velocity field (e.g., Paper Way Bar if the Sun is placed just outside the 2:1 Outer I) about the LSR and write the mean radial velocity vd Lindblad Resonance (OLR). Using high-resolution spec- and longitudinal proper motion µl as functions of the tra of nearby F and G dwarf stars, Bensby et al. (2007) Galactic longitude l as have investigated the detailed abundance and age struc- v ture of the Hercules stream. Since this stream is com- d = K + A sin(2l)+ C cos(2l) (1) posed of stars of different ages and metallicities pertinent d to both thin and thick disks, they concluded that a dy- µl = B + A cos(2l) − C sin(2l) namical effect, such as the influence of the Bar, is a more where d is the average heliocentric distance of stars, A likely explanation than a dispersed cluster. and B are the usual Oort constants, and C and K are given by 1 Department of Physics and Astronomy, University of u ∂u 1 ∂v Rochester, Rochester, NY 14627; [email protected], 2C ≡− + − φ (2) [email protected] r ∂r r ∂φ 2 Laboratory for Laser Energetics, University of Rochester, u ∂u 1 ∂v Rochester, NY 14623; [email protected] 2K ≡ + + + φ . (3) r ∂r r ∂φ 2

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

Here r and φ are the usual polar coordinates and vφ = v0+v, where v0 is the circular velocity at the Solar radius, r0. In this work we primarily consider a flat rotation curve (RC) (see, however §5), hence the derivatives of vφ in the above equations are identical to the derivatives of v. C describes the radial shear of the velocity field Fig. 1.— Simulated u − v distribution for a Bar pattern speed ◦ and K its divergence. For an axisymmetric Galaxy we Ωb = 1.87Ω0, Bar angle φ0 = 35 and a sample depth d = r0/40. expect vanishing values for both C and K 3. Whereas C The initial velocity dispersion is σu = 0.18v0 and the Bar strength is ǫb = −0.012. Contour levels are equally spaced. The clump iden- could be derived from both radial velocities and proper tified with the Hercules stream is clearly discernible in the lower motions, K can only be measured from radial velocities, left portion of the plot. This figure is in agreement with the ob- in which case accurate distances are also needed. served f0(u, v) (Dehnen 1998; Fux 2001) and simulated distribution A problem with using proper motions data has been functions of Dehnen (2000) and Fux (2001). potential due to the disk and halo has the form Φ (r)= described by O&D. The authors present an effect which 0 v2 log(r), corresponding to a flat RC. arises from the longitudinal variations of the mean stel- 0 lar parallax caused by intrinsic density inhomogeneities. 3.1. The Bar potential Together with the reflex of the solar motion these varia- tions create contributions to the longitudinal proper mo- We model the nonaxisymmetric potential perturbation tions which are indistinguishable from the Oort constants due to the Galactic Bar as a pure quadrupole 3 at ≤ 20% of their amplitude. O&D corrected for the rb , r ≥ r “mode-mixing” effect described above, using the latitu-  r b Φb = Ab(ǫb) cos[2(φ − Ωbt)] ×  3 dinal proper motions. The resulting C is found to vary  2 − r , r ≤ r  rb  b approximately linearly with both color and asymmetric  (4) drift (and thus mean age) from C ≈ 0 km/s/kpc for Here A (ǫ ) is the Bar’s gravitational potential ampli- blue samples to C ≈ −10 km/s/kpc for late-type stars b b tude, identical to the same name parameter used by (see Figs. 6 and 9 in O&D). Since C is related to the Dehnen (2000); the strength is specified by ǫ = −α radial streaming of stars, we expect non-axisymmetric b from the same paper. The Bar length is r = 0.8r structure to mainly affect the low-dispersion population b cr with r the Bar corotation radius. The pattern speed, which would result in the opposite behavior for C. In cr Ω is kept constant. We grow the Bar by linearly vary- Paper I we showed that spiral structure failed to explain b ing its amplitude, ǫ , from zero to its maximum value the observed trend of C. b in four Bar rotation periods. We present our results by Note that the Oort constants are not constant unless changing Ω and keeping r fixed. The 2:1 outer Lind- they are measured in the SN. Due to nonaxisymmetries b 0 blad resonance (OLR) with the Bar is achieved when they may vary with the position in the Galaxy. Thus, the Ω /Ω = 1+ κ/2 ≈ 1.7, where κ is the epicyclic fre- Oort constants have often been called the Oort functions. b 0 quency. We examine a region of parameter space for a 3. THE SIMULATIONS range of pattern speeds placing the SN just outside the OLR. For a given pattern speed one could obtain the We perform 2D test-particle simulations of an initially ratio r0/rOLR through r0/rOLR =Ωb/ΩOLR ≈ Ωb/1.7. axisymmetric exponential galactic disk. To reproduce In contrast to Dehnen (2000) and similar to Fux (2001) the observed kinematics of the Milky Way, we use disk and M¨uhlbauer & Dehnen (2003), we integrate forward parameters consistent with observations (see Table 1). in time. To allow for phase mixing to complete after A detailed description of our disk model and simulation the Bar reaches its maximum strength, we wait for 10 technique can be found in Paper I. We are interested in Bar rotations before we start recording the position and the variation of C with color (B−V ), and asymmetric velocity vectors. In order to improve statistics, positions drift va. We simulate variations with color by assuming and velocities are time averaged for 10 Bar periods. After that the velocity dispersion increases from blue to red utilizing the two-fold symmetry of our galaxy we end star samples. For a cold disk we start with an initial up with ∼ 5 × 105 particles in a given simulated solar radial velocity dispersion σu = 0.05v0 whereas for a hot neighborhood with maximum radius dmax = r0/10. disk we use σu =0.18v0. The background axisymmetric 4. RESULTS 3 Note, however, that C and K would also be zero in the presence of nonaxisymmetric structure if the Sun happened to be located on First we show that we can reproduce the results of a symmetry axis. Dehnen and Fux for the Hercules stream. We present a 3

Fig. 2.— Each panel shows the variation of the Oort constant C with Bar angle φ0, for a simulation with the parameters given in Table 1 and a particular Bar pattern speed, Ωb, and a mean sample depth, d. Solid and dotted lines correspond to cold- and hot-disk values, respectively. Columns from left to right, show an increasing Ωb in units of Ω0. Note that the OLR is at ΩOLR ≈ 1.7. Different rows present results from samples with different mean heliocentric distance, d. Good matches to the observed trend in C (vanishing value for the cold ◦ ◦ disk and a large negative for the hot one) are achieved for 20 6 φ0 6 45 and 1.83 6 Ωb/Ω0 6 1.91. simulated u − v velocity distribution for Ωb = 1.87Ω0, Ch (dotted lines in Fig. 3.1) varies with galactic az- ◦ φ0 = 35 , and a sample depth d = r0/40 in fig. 1. The imuth as Ch(φ0) ∼ sin(2φ0) for all of the Ωb values con- initial velocity dispersion is σu =0.18v0. This plot is in- sidered. On the other hand, the cold disk values (solid deed in very good agreement with previous test-particle line) exhibit different variations, depending on Ωb or (Dehnen 2000; Fux 2001) and N-body (Fux 2001) simu- equivalently, on the ratio r0/rOLR. Closer to the OLR lations. (left columns of Fig. 3.1), Cc(φ0) approaches the func- By a quantitative comparison of the observed with tional behavior of Ch(φ0). Away from the OLR Cc(φ0) ◦ the simulated distributions, Dehnen (2000) deduced the is shifted by 90 compared to Ch(φ0). While both cold Milky Way Bar pattern speed to be Ωb/Ω0 =1.85±0.15. and hot disks yield in increase in the amplitude of C(φ0) In this calculation only the local velocity field, i.e., the as the pattern speed nears the OLR, the effect on the observed u−v distribution, was taken into account. Now, Cc(φ0) is much stronger. This is consistent with our if one could also relate the dynamical effect of the Bar to expectation that the cold disk is affected more by the the derivatives of the velocities, an additional constraint Bar, especially near the OLR. While close to the OLR on Bar parameters would be provided. Since the gradi- |Ch(φ0)| < |Cc(φ0)|, we observe the opposite behavior ents of u and v are hard to measure directly, one needs away from it. This could be explained by the results of an indirect way of achieving this task. The obvious can- M¨uhlbauer & Dehnen (2003), where they find that high didates, as anticipated, are the Oort constants C and velocity dispersion stars tend to shift the “effective reso- K. Oort’s K, however, is hard to measure as mentioned nance” radially outwards. above; hence only C, as estimated by O&D, will be em- According to O&D’s “mode-mixing” corrected value, ployed here. By Fourier expansion of eqs. 1 (see Paper C ≈ 0 for the cold sample and decreases to about -10 I for details) we estimate C from our numerically simu- km/s for the hot population. Hence we look for loca- lated velocity distributions for the initially cold and hot tions in Fig. 3.1 complying with this requirement. In disks. the transition region, 1.83 6 Ωb/Ω0 6 1.91, where the function Cc(φ0) transforms into its negative, there are 4.1. Variation of C with Bar pattern speed and specific angles which provide good matches for the ob- orientation servations, i.e., Cc(φ0) ≈ 0 while Ch(φ0) is significantly negative. We performed simulations with different pat- In Fig. 3.1 we present our results for C as a function tern speeds in the range 1.7 6 Ωb/Ω0 6 2.5. It was of the Bar angle, φ0 (the angle by which the Sun’s az- found that only in the transition region in Fig. 3.1 could imuth lags the Bar’s major axis). Each column shows one achieve a satisfactory match to the observations. Al- a simulation with a different pattern speed, indicated in though the left- and right-most columns are nearly con- each plot. Rows from top to bottom show C as calcu- sistent with our requirement on C, we reject these values lated from samples at average heliocentric distances cor- of Ωb for the following reason. O&D estimated a large responding to d = 200, d = 400, and d = 600 pc, for a negative C from both the reddest main-sequence stars, Solar radius r0 = 7.8 kpc. Solid and dotted lines repre- which are nearby, and the more distant red giants. This sent the results for cold and hot disks, respectively. The is inconsistent with the largest mean sample depth shown dashed lines indicates C = 0. C is presented in units of in Fig. 3.1 with Ωb = 1.81, 1.93. We conclude that the Ω0 = v0/r0. To make the discussion less cumbersome, we Bar pattern speed must lie in the range write Ch and Cc to refer to the values for C as estimated from the hot and cold disks, respectively. Ωb/Ω0 =1.87 ± 0.04. (5) 4

The Bar angle with respect to the Sun has been pro- for Ωb lies well within the estimate by Dehnen (2000) posed, as derived from IR photometry, to lie in the range and that by Debattista et al. (2002), based on OH/IR 15◦ − 45◦ with the Sun lagging the Bar major axis, star kinematics. This study provides an improvement on whereas Dehnen (2000) found that the Bar reproduced the measurement of the Bar pattern speed by a factor of ◦ ◦ the u-anomaly for 10 6 φ0 6 70 . Examining Fig. 3.1, ∼ 4 compared to previous work (Dehnen 2000). The im- we find that our requirements on C result in constraining proved constraints on Bar parameters should be tested ◦ ◦ the Bar angle in the range 20 6 φ0 6 45 , depending by, and will benefit, future studies of the Bar structure on Ωb. This result for the dependence of pattern speed in the Galactic center region, that will become possi- on Bar angle is in a very good agreement with Fig. 10 ble with future radial velocity and proper motion studies by Dehnen (2000) which shows a linear increase of the (e.g., BRAVA, Rich et al. 2007). derived Ωb/Ω0 as a function of φ0. This is remarkable In addition to a flat RC we have also considered a β since Dehnen (2000) obtained his results in a completely power law initial tangential velocity vφ = v0(r/r0) with different way than our method here. β = 0.1, −0.1 corresponding to a rising and a declining RC, respectively. We found that for β = 0.1 no addi- 4.2. The effect of the Bar on A and B tional error in Ωb is introduced. However, for β = −0.1 The Oort constants A and B were found to also be we estimated Ωb/Ω0 =1.85 ± 0.06. From Fig. 3.1 we see affected by the Bar, although to a much smaller ex- that the main source of error in the Bar pattern speed is tent. One could estimate the Bar-induced errors as due to the uncertainty in the Bar angle. A recent work by Rattenbury et al. (2007) used OGLE-II microlensing ∆A ≡ A − Aaxi and ∆B ≡ B − Baxi. Here Aaxi, Baxi are observations of red clump giants in the Galactic bulge, the resulting axisymmetric values when simulations ex- ◦ ◦ cluding the Bar perturbation are performed. Note that to estimate a bar angle of 24 − 27 . Using this as an we require a different simulation with each of the initial additional constraint we obtain Ωb/Ω0 =1.84 ± 0.01. radial velocity dispersions used in order to estimate the While we account for the trend of increasingly nega- asymmetric drift induced errors. We found that for the tive C with increasing velocity dispersion, we fail to re- cold disk the Bar causes ∆A =3.5±0.7 and ∆B =3±0.6 produce the size of the C-value measured by O&D. The km/s/kpc, while the hot disk yielded ∆A =0.5±1.5 and most negative value we predict is about -6 while they ∆B = 0 ± 1 km/s/kpc. These values are calculated for measure -10 km/s/kpc. We discuss possible reasons for this discrepancy: (i) we expect that the magnitude of r0 =7.8 kpc and v0 = 220 km/s. Despite the Bar’s effect on C, the hot population does provide better measure- C is related to the fraction of SN stars composing the ments for A and B. Hercules stream. It is possible that during its formation O&D found an approximately linear variation of A and the Bar changed its pattern speed forcing more stars to B with color with A increasing and B decreasing. The be trapped in the 2:1 OLR, thus increasing this frac- Bar induced errors presented here match the observed tion. This would give rise to a more negative C from hot trend of B but have the opposite one for A. It is possible stars, while leaving the cold population unaffected; (ii) that local spiral structure is responsible for this discrep- the numerical model here uses a distance limited sample. ancy. Modeling a stellar population with a magnitude limited sample is more appropriate to compare to the observed 5. CONCLUSION measurements. (iii) O&D’s “mode-mixing” correction is We have shown that the Galactic Bar can account for only valid if the Sun’s motion in the z-direction were as- a trend seen in measurements of Oort’s C, namely a sumed constant. This is invalidated if, for example, a more negative C-value with increasing velocity disper- local Galactic warp were present. sion. This trend is difficult to explain with other dy- namical models (e.g., Paper I). By requiring that the Bar model accounts for this trend in C, we have im- Support for this work was in part provided proved the measurement of the Bar pattern speed, find- by NSF grant ASST-0406823, and NASA grant ing Ωb/Ω0 = 1.87 ± 0.04. In addition, the Bar angle is No. NNG04GM12G issued through the Origins of Solar ◦ ◦ found to be in the range 20 6 φ0 6 45 . Our result Systems Program.

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