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Schematic view of the equatorial X-Y plane of a four-arm spiral barred galaxy in the adopted reference frame. The main perturbation components, such as the spiral arms, the central bar and bulge, were calculated using the parameter's values from Table 1, except for the pitch angle i = +14 •. The position of the Sun is shown by a blue cross. The bar's phase with respect to the reference direction (X-axis) is 67 • .5 (or 22 • .5 with respect to the Sun). Close to the Sun, Sagittarius and Perseus arms are identified.
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Context . Observational data indicate that the Milky Way is a barred spiral galaxy. Computation facilities and availability of data from Galactic surveys stimulate the appearance of models of the Galactic structure, however further efforts are needed to build dynamical models containing both spiral arms and the central bar/bulge.
Aims . We expand t...
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... and (7) adopted in this work, and their physical meanings are given in Table 1 (the detailed discussion on this choice can be found in Papers I and II). Figure 1 shows, by black curves, the loci of four arms on the (X = R cos ϕ, Y = R sin ϕ)-plane, which were obtained as az- imuthal minima of the potential (Eq. 6) in the reference frame defined as follows. The origin of the reference frame lies at the Galactic centre, while the equatorial plane of the Galaxy is de- fined as a reference plane. ...
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... (Eq. 6) in the reference frame defined as follows. The origin of the reference frame lies at the Galactic centre, while the equatorial plane of the Galaxy is de- fined as a reference plane. The axis X (ϕ = 0) is fixed in such a way that the Sun's azimuthal coordinate is ϕ = 90 • , placing the Sun on the Y-axis at R = 8.0 kpc (blue cross in Fig. 1). The orientation of the spiral arms on the X-Y plane is fixed by the value of the free parameter γ in Eq. (7). We choose γ-value such that the Sun (located at R = 8.0 kpc and ϕ = 90 • ) is 1 kpc from the Sagittarius arm locus. Thus we obtain γ = 237 • .25 for the spirals parameters from Table ...
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... Figure 1 shows the projection of the central bar on the equa- torial plane as a very eccentric ellipse, with the semi-major axis R bar = 2.9 kpc and the initial phase of γ 0 bar = 67 • .5, with respect to the X-axis. This value of γ 0 bar places the bar's semi-major axis at an angle of 22 • .5 with respect to the direction Sun-Galactic centre. ...
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... Fig. 1 also shows the projection of the central bulge on the galactic equatorial plane as a circle. The radius of the cir- cle is assumed to be 1 kpc and the polar flattening f bulge = 0.2. The potential of the oblate ellipsoid that describes the central bulge introduces no asymmetric perturbations to the star's mo- tion on the equatorial ...
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... of the bar, we plot the energy levels in thick black lines in Fig. 4. The zone of influence extends up to ∼3.5 kpc along the main axis of the bar, that is, beyond the phys- ical extension of the bar with R bar = 2.9 kpc (see Table 1). The geometry of the zone of influence also differs from that of an elongated ellipsoid which shapes the bar (see Fig. 1); it is rather like an observable box-shaped ...
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... the levels in grey lines in Fig. 4. In the intermediate zone, the loci of the azimuthal minima of the potential H 1 (R, ϕ) (blue curves), which, outside the zone of influence of the bar, correspond to the loci of the main spiral arms in our model, are strongly perturbed; this is noted from the comparison with the unperturbed spiral arms shown in Fig. 1. The two arms connected with the extremes of the bar, where the bar's force function is strongest, suffer significant deformation, while the other two arms are partially vanishing. We estimate that the intermediate zone extends up to ∼4.3 kpc until its effects disappear. It is interesting to note that the upper boundary of the ...
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... understand the stellar dynamics in this case, we integrate the orbit of the Sun using two different values of M bar , 1 × 10 9 (red) and 3 × 10 10 (blue) solar masses, keeping the other pa- rameters from Table 1; the trajectories obtained are shown in Fig. 10 by red and blue points, respectively. Both orbits, starting at the same initial configuration (a blue cross symbol), are librat- ing; however, the red path librates around the L 4 -centre of the spiral corotation, while the blue path librates around B 1 -centre of the bar's corotation. The first orbit oscillates between the ...
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... functions were calculated along the major axis of the bar fixed at ϕ = 67 • .5 (see Sect. 2.3), where the bar's perturbation is strongest; the parameters were taken from Table 1. Figure 11 shows the families as functions of the Galactocen- tric distance by black curves; for the sake of comparison, we also plot the force function of the spiral arms by a red curve. We note that the non-axisymmetric perturbation of the bar be- comes stronger with the increasing flattening of the bar, f bar . ...
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... Fig. 11 shows that, for M bar = 10 9 M , the zone of influence of the bar never approaches the domain of the spiral corotation, where the Sun is evolving. Therefore, it is expected that, for the adopted bar's mass, the motion of the Sun and the lo- cation of the L 4 -centre on the X-Y plane is only slightly affected by increasing f bar ...
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... test the effects of the bar's size on the Sun's motion, which is representative of the dynamical stability of objects from the local arm. For this, we construct the dynamical map on the parametric plane M bar -R bar shown in Fig. 12. Varying the bar's mass in the range between 10 8 M and 1.5×10 10 M and the bar's radius between 1 kpc and 6 kpc, we analyse the dynamical sta- bility of the Sun, with coordinates X = 0 and Y = 8.0 kpc, and the velocities p R = −11 km s −1 and V θ = 242.24 km s −1 . The rest of the parameters are taken from Table ...
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... perturbations to stellar motion due to the bar are weak in the light-tone domain on the dynamical map in Fig. 12; they in- crease in the darker zones and provoke strong instabilities in the red-hatched region. (The fine effects in the solar motion shown by slight variations of grey tones in the domain of stable orbits are not analysed here.) The analysis of the map shows the ef- fects of the mass of the bar and of its radius on the stability of ...
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... a physically plausible model, the observable structures must be stable over at least a few billion years (see discussion in Paper II). There are two physical structures present in our model which should remain stable over this period: the local arm, lo- Fig. 14. Dynamical map on the parametric plane Ω bar -M bar for the orbit with initial conditions R = R bar , ϕ = γ 0 bar , p R = 0 and V θ = V rot (R bar ), representative of the bar structure. The light grey tones represent reg- ular orbits, while increasingly dark tones correspond to increasing in- stabilities and chaotic motion. The ...
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... top of Fig. 13 shows the dynamical map for the L 4 - centre on the parametric plane Ω bar -M bar . The increasingly dark tones indicate the appearance of dynamical instabilities and strong chaotic motions (red hatched regions). which is varied between 15 and 70 km s −1 kpc −1 . For the bar's corotation resonance, for instance, the condition Ω p ≈ Ω ...
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... and strong chaotic motions (red hatched regions). which is varied between 15 and 70 km s −1 kpc −1 . For the bar's corotation resonance, for instance, the condition Ω p ≈ Ω bar (but not Ω p ≡ Ω bar ) will produce the overlap with the spiral coro- tation zone and, consequently, generate dynamical instabilities. This is what we observe in Fig. 13, where the domains (in darker tones) surrounding the nominal position of the main resonances are ...
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... this stable behaviour, we associate to the bar an orbit which starts at its near extremity, with initial conditions R = R bar , ϕ = γ 0 bar , p R = 0 and V θ = V rot (R bar ). We consider the stability of this orbit as an indicator of the bar's sta- bility. The dynamical map for this orbit on the parametric plane Ω bar − M bar is shown in Fig. 14. We see that, for all values of Ω bar , the orbit is stable only for M bar < 10 10 M , for low Ω bar . More- over, for high Ω bar , a bar's mass of ∼ 10 9 M leads to a high de- gree of instability for the orbit. For this range of masses, stability of this orbit imposes an upper limit of Ω bar < 50 km s −1 kpc −1 ...
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... done here is sufficient to constrain the bar's mass to M bar ∼ 2 × 10 9 M and the bar's angular velocity to Ω bar < 50 km s −1 kpc −1 . This leads to a situation wherein it is unlikely that the bar's OLR lies near the solar radius, since it should then have Ω bar ≈ 47 km s −1 kpc −1 , which is close to the upper acceptable limit for Ω bar (see Fig. 14). The bar's OLR is most probably outside the solar radius. Moreover, Fig. 13 corroborates this conclusion. The chaotic region originated by the resonance overlap in the local corotation zone, for Ω bar ≈ 47 km s −1 kpc −1 , would be an obsta- cle for the formation of the local arm (see Paper ...
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... and the bar's angular velocity to Ω bar < 50 km s −1 kpc −1 . This leads to a situation wherein it is unlikely that the bar's OLR lies near the solar radius, since it should then have Ω bar ≈ 47 km s −1 kpc −1 , which is close to the upper acceptable limit for Ω bar (see Fig. 14). The bar's OLR is most probably outside the solar radius. Moreover, Fig. 13 corroborates this conclusion. The chaotic region originated by the resonance overlap in the local corotation zone, for Ω bar ≈ 47 km s −1 kpc −1 , would be an obsta- cle for the formation of the local arm (see Paper ...
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... model of the bar predicts the existence of four regions of complicated geometry around it: the zig-zags in the grooves which represent the minima of gravitational potential, at the ex- tremities of the bar, and the short sectors of spiral arms, at 90 • from the main axis. We numbered these regions from 1 to 4 in Fig. 15 by green crosses; they allow us to impose restrictions on the size and orientation of the bar. These masers are associated with massive star-formation regions and molecular clouds. Observations of external barred galaxies tell us that there is no star formation inside bars (James & Percival 2018). Therefore, the abrupt cut in the space ...
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... which is consistent with their proximity to the Galactic centre. We know, from the experience with the local arm discussed in Paper II, that zones of resonances also harbour regions of star formation, with the presence of masers. The coincidence of the two masers with the small-sized zone of potential maximum (as indicated by the contour lines in Fig. 15) seems to establish a strong restriction on the inclination of the bar, since a small ro- tation of the bar would destroy the coincidence, as well as a re- striction on the width of the ...
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... at a longitude of −22 • (or 338 • ) with a cumulation of masers with a range of velocities that reaches −80 km s −1 . The authors attribute this feature to the Perseus arm origin. We note that the segment of arm in region 4 (with nega- tive potential, indicated in blue) is not the origin of the Perseus arm, in our model. However, we can see from Fig. 15 that it is situated in the direct prolongation of the Perseus arm, and could be interpreted as being part of it. The observations of Green et al. reinforce our interpretation that region 4, situated at a longitude of −22 • , is a region containing methanol ...
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... cited studies obtained putting the bar's OLR near the Sun's position. Indeed, we verified, from a dynamical map of the solar neighbourhood on the U-V plane, that a model accounting solely for the bar and bulge perturbations (with the values of the parameters taken from Table 1) does not produce a Hercules- stream-like feature. In the top panel of Fig. 16, we show the dy- namical map of the U-V plane calculated for such a model. On the other hand, a model accounting for the spiral arms perturbation produces more interesting features in the dynami- cal map of the U-V plane. The bottom panel of Fig. 16 shows the iso-density contours of stars in the observed U-V plane of the so- lar ...
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... taken from Table 1) does not produce a Hercules- stream-like feature. In the top panel of Fig. 16, we show the dy- namical map of the U-V plane calculated for such a model. On the other hand, a model accounting for the spiral arms perturbation produces more interesting features in the dynami- cal map of the U-V plane. The bottom panel of Fig. 16 shows the iso-density contours of stars in the observed U-V plane of the so- lar neighbourhood, taken from data of the Geneva-Copenhagen survey catalogue ( Holmberg et al. 2009), superposed to the dy- namical map of the modelled U-V plane. In this case, the model includes the perturbation from the spiral arms and from the bar/bulge ...
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... Coma Berenices) lie, approximately, inside the spiral corotation zone. We see a clear correlation between the observed structures and this resonance (in the central region of the plane). Apart from the relationship between the main moving groups and the coro- tation zone, we focus next on the Hercules stream and the chains of resonances seen in Fig. ...
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... propose this resonance as a likely dynamical origin of the Hercules stream, given the prox- imity between the extensions of these two features in the U-V plane. At first sight, a connection between these structures is not particularly evident; this is due to the fact that the orbits for the construction of the dynamical maps on the U-V plane in Fig. 16 were integrated fixing the initial conditions of the test-particle at R = 8 kpc and ϕ = 90 • , and varying the U and V velocity values. As a consequence, these conditions influence the aspects and positions of the chains of resonances in the U-V plane. In fact, if we fix the initial U-value of the orbits at −28 km s −1 , the 8/1 ...
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... i is the spiral pitch angle; R i is a reference radius and γ is an arbitrary phase angle, whose values define the orientation of the spirals in the chosen reference frame. The values of the parameters in Eqs. (6) and (7) adopted in this work, and their physical meanings are given in Table 1 (the detailed discussion on this choice can be found in Papers I and II). Figure 1 shows, by black curves, the loci of four arms on the (X = R cos ϕ, Y = R sin ϕ)-plane, which were obtained as azimuthal minima of the potential (6) in the reference frame de- fined as follows. The origin of the reference frame lies at the Galactic center, while the equatorial plane of the Galaxy is de- fined as a reference plane. The axis X (ϕ = 0) is fixed in such a way that the Sun's azimuthal coordinate is ϕ = 90 • , placing the Sun on the Y-axis at R = 8.0 kpc (blue cross in Figure 1). The orientation of the spiral arms on the X-Y plane is fixed by the value of the free parameter γ in the expression (7). We choose γ-value such that the Sun (located at R = 8.0 kpc and ϕ = 90 • ) is 1 kpc distant from the Sagittarius arm locus. Thus we obtain γ = 237 • .25, for the spirals parameters from Table ...
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... i is the spiral pitch angle; R i is a reference radius and γ is an arbitrary phase angle, whose values define the orientation of the spirals in the chosen reference frame. The values of the parameters in Eqs. (6) and (7) adopted in this work, and their physical meanings are given in Table 1 (the detailed discussion on this choice can be found in Papers I and II). Figure 1 shows, by black curves, the loci of four arms on the (X = R cos ϕ, Y = R sin ϕ)-plane, which were obtained as azimuthal minima of the potential (6) in the reference frame de- fined as follows. The origin of the reference frame lies at the Galactic center, while the equatorial plane of the Galaxy is de- fined as a reference plane. The axis X (ϕ = 0) is fixed in such a way that the Sun's azimuthal coordinate is ϕ = 90 • , placing the Sun on the Y-axis at R = 8.0 kpc (blue cross in Figure 1). The orientation of the spiral arms on the X-Y plane is fixed by the value of the free parameter γ in the expression (7). We choose γ-value such that the Sun (located at R = 8.0 kpc and ϕ = 90 • ) is 1 kpc distant from the Sagittarius arm locus. Thus we obtain γ = 237 • .25, for the spirals parameters from Table ...
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... R bar is the semi-major axis (or radius) of the central bar, m = 4 and f m is the shape function given in Eq. (7). From obser- vations, the current bar's phase with respect to the Sun varies in the range from 10 • to 30 • (that is, from 60 • to 80 • with respect to the X-axis) ( Bobylev et al. 2014); thus, using the above expres- sion, we can assume the bar's radius in the range from 2.78 kpc to 3.05 kpc (see details in Sect. 2.4.3). Figure 1 shows the projection of the central bar on the equa- torial plane as a very eccentric ellipse, with the semi-major axis R bar = 2.9 kpc and the initial phase of γ 0 bar = 67 • .5, with respect to the X-axis. This value of γ 0 bar places the bar's semi-major axis at the angle of 22 • .5 with respect to the direction Sun-Galactic center. The eccentricity of the ellipse is defined by the bar's flat- tening, whose starting value is chosen as f bar = 0.7 (see Table ...
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... Figure 1 also shows the projection of the central bulge on the galactic equatorial plane as a circle. The radius of the circle is assumed to be 1 kpc and the polar flattening f bulge = 0.2. The potential of the oblate ellipsoid which describes the central bulge introduces no asymmetric perturbations to the star's motion on the equatorial plane. On the other hand, the ax- isymmetric perturbations are already included in the unperturbed potential Φ 0 (R) via the rotation curve, as discussed in Sect. 2.1. Therefore, the bulge's contribution, Φ bulge (R), in the Hamiltonian (2), can be disconsidered in the case when the stellar motion is confined to the equatorial plane. Despite this fact, in this paper, we will consider the term Φ bulge (R) always together with the term Φ bar (R, ϕ, t), referring to them as a bar/bulge perturbation. The application of the model of the bar/bulge requires the knowl- edge of the parameters of this structure. From these, we paid a special attention to the choice of the total mass, rotation speed and size of the bar. Indeed, as will be shown below, the values of these parameters are crucial for the stability of solar motion inside the corotation zone, the stable corotation island encom- passing the Sun (see Lépine et al. ...
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... us now consider the two masers observed by Sannaet al. (2014), one H 2 O maser with longitude l = 10 • .472 and distance d = 8.55 kpc with respect to the Sun, and the other a methanol maser with l = 12 • .025 and d = 9.43 kpc. Both appear to coin- cide with the ILR, in a direction where the influence of the bar is minimal. The parallax measurements of these two masers are quite accurate, with errors about 0.008 mas. The LSR velocities of the two sources are high (69 and 108 km s −1 , respectively), which is consistent with their proximity to the Galactic center. We know, from the experience with the Local Arm discussed in Paper II, that zones of resonances also harbour regions of star formation, with the presence of masers. The coincidence of the two masers with the small-sized resonance zone (as indicated by the contour lines in Figure 15) seems to establish a strong re- striction on the inclination of the bar, since a small rotation of the bar would destroy the coincidence, as well as a restriction on the width of the ...
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... the symmetric positive potential zone on the other side of the bar (region 4), we did not find any report of VLBI ob- servations of masers, but interesting data are available in litera- ture. Green et al. (2011) investigated the distribution of methanol maser sources in the inner Galaxy, and found the most prominent tangential direction at longitude −22 • (or 338 • ) with a cumula- tion of masers with a range of velocities that reaches −80 km s −1 . The authors attribute this feature to the Perseus arm origin. We note that the segment of arm in region 4 (with negative poten- tial, indicated in blue) is not the origin of the Perseus arm, in our model. However, we can see from Figure 15 that it is situ- ated in the direct prolongation of the Perseus arm, and could be interpreted as being part of it. The observations of Green et al. reinforce our interpretation that region 4, situated at longitude −22 • , is a region containing methanol ...
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... the other hand, a model accounting for the spiral arms perturbation produces more interesting features in the dynamical map of the U-V plane. The bottom panel of Figure 16 shows the iso-density contours of stars in the observed U-V plane of the so- lar neighbourhood, taken from data of the Geneva-Copenhagen survey catalogue ( Holmberg et al. 2009), superimposed on the dynamical map of the modeled U-V plane. In this case, the model includes the perturbation from the spiral arms and from the bar/bulge ...
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... that observable objects avoid the domains of high instabilities, we can deduce the constraint on the rotation speed of the bar: its value must lie outside the zones of influence of the strong low-order resonances on the parametric plane shown in Figure 13 top. However, as shown in all previous sections, the situation is different when the bar's speed matches exactly the pattern speed; in this case, the spiral arms and the bar/bulge form an unique structure whose origin would still need to be ex- ...
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... the aforementioned results, we would not expect that our bar model, with a pattern speed of 28.5 km s −1 kpc −1 , plac- ing the bar's corotation in the vicinity of the solar radius, could generate a bimodal feature in the local velocity space in the same way as the cited studies obtained putting the bar's OLR near the Sun's position. Indeed, we verified, from a dynamical map of the solar neighbourhood on the U-V plane, that a model accounting solely for the bar and bulge perturbations (with the values of the parameters taken from Table 1) does not produce a Hercules- stream-like feature. In the top panel in Figure 16, we show the dynamical map of the U-V plane calculated for such ...
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... a physically plausible model, the observable structures must be stable over at least a few billion years (see discussion in Paper II). There are two physical structures present in our model which should remain stable over this period: the Local Arm, lo- cated near the solar radius, and the Galactic bar. Let us first con- sider the Local Arm. Since it is associated with the spiral coro- tation zone (see Paper II), the stability of this zone is important for preserving the Local Arm structure. We analyze the stability of the spiral corotation zone as a function of the parameters Ω bar and M bar . We consider in this section, as representative of this zone, the orbit of the L 4 -center calculated with the parameters of the basic model (see Table 1). Perturbations due to different angular speeds of the bar structure affect the orbit of this point, which can even become chaotic. In non-linear dynamics studies, this phenomenon is known as an overlap of resonances (see details in Lichtenberg & Lieberman 1992). It happens when two (or more) distinct resonances are sufficiently close to each other in a phase space and, con- sequently, their overlap results in the appearance of widespread (large-scale) chaos. In our case, there are two distinct sources of resonances: the spiral arms rotating with the pattern speed Ω p = 28.5 km s −1 kpc −1 , and the bar rotating with the speed Ω bar , which varies between 15 and 70 km s −1 kpc −1 . For the bar's coro- tation resonance, for instance, the condition Ω p ≈ Ω bar (but not Ω p ≡ Ω bar ) will produce the overlap with the spiral corotation zone and, consequently, generate dynamical instabilities. This is what we observe in Figure 13 top, where the domains (in darker tones) surrounding the nominal position of the main resonances are ...
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... more prominent island of stability in the bottom part of the dynamical map, with V < −50 km s −1 , is associated with the 8/1 resonance of the spiral pattern. Here we propose this reso- nance as a likely dynamical origin of the Hercules stream, given the proximity between the extensions of these two features in the U-V plane. At a first sight, a connection between these struc- tures could not be too evident; this is due to the fact that the orbits for the construction of the dynamical maps on the U-V plane in Figure 16 were integrated fixing the initial conditions of the test-particle at R = 8 kpc and ϕ = 90 • , and varying the U and V velocity values. As a consequence, these conditions in- fluence the aspects and positions of the chains of resonances in the U-V plane. In fact, we verified that, if we fixed the initial V-value of the orbits at −50 km s −1 , the 8/1 resonance island could extend from U = −60 km s −1 to U = 20 km s −1 (cen- tered at V = −50 km s −1 ), thus better matching the position of the Hercules stream in the observed U-V plane. Also, small dis- placements of the initial radius, for example R = 7.8 or 7.9 kpc, produce good matches between the 8/1 resonance and Hercules stream. We also verified that several orbital trajectories of stars in the Hercules stream present a radial oscillation compatible with being inside a 8/1 resonance or quasi-resonance with the spiral pattern. Minor contributions from the 9/1 and 10/1 resonances may also be ...
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... summary, for order-of-magnitude estimates, the analysis done here is sufficient to constrain the bar's mass to M bar ∼ 2 × 10 9 M and the bar's angular velocity to Ω bar < 50 km s −1 kpc −1 . It results that it is unlikely that the bar's OLR lies near the solar radius, since it should have then Ω bar ≈ 47 km s −1 kpc −1 , which is close to the upper acceptable limit for Ω bar . The bar's OLR is most probably outside the solar radius. Moreover, Figure 13 top corroborates this conclusion. The chaotic region originated by the resonance overlap in the local corotation zone, for Ω bar ≈ 47 km s −1 kpc −1 , would be an obstacle for the formation of the Local Arm (see Paper ...
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... physical structure which must also be stable is the bar itself. In order to quantify this stable behaviour, we associate to the bar an orbit which starts at its near extremity, with initial conditions R = R bar , ϕ = γ 0 bar , p R = 0 and V θ = V rot (R bar ). We consider the stability of this orbit as an indicator of the bar's stability. The dynamical map for this orbit on the parametric plane Ω bar − M bar is shown in Figure 14. We see that, for all values of Ω bar , the orbit is stable only for M bar < 10 10 M , for low Ω bar . Moreover, for high Ω bar , a bar's mass of ∼ 10 9 M leads to a high degree of instability for the orbit. For this range of masses, stability of this orbit imposes a an upper limit of Ω bar < 50 km s −1 kpc −1 ...
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... main moving groups (Pleiades, Hyades, Sirius, and Coma Berenice) lie, approximately, inside the spiral corotation zone. We see a clear correlation between the observed structures and this resonance (in the central region of the plane). Apart from the relationship between the main moving groups and the coro- tation zone, we focus next on the Hercules stream and the chains of resonances seen in Figure 16 ...
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... bar's flattening is simple to be analysed. According to the definition in Eqs. (8), its possible values lie between 0 and 1. We calculate the families of the force function of the bar parameter- ized by the different values of f bar , from 0.1 to 0.9. All functions were calculated along the major axis of the bar fixed at ϕ = 67 • .5 (see Sect. 2.3), where the bar's perturbation is strongest; the pa- rameters were taken from Table 1. Figure 11 shows the families as functions of the Galactocen- tric distance by black curves; for the sake of comparison, we also plot the force function of the spiral arms by a red curve. We note that the non-axisymmetric perturbation of the bar be- comes stronger with the increasing flattening of the bar, f bar . The domain of the overlap with the spiral perturbation is also increasing, which means that the inner and intermediate zone of the influence of the bar (see Sect. 3) are expanding. Indeed, for f bar > 0.5, the force function of the bar dominates over the spi- rals, even beyond its physical extension given by R bar = 2.9 ...
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... distribution of masers with accurate positions measured with VLBI (see the Bessel Survey 4 ), are displayed in Figure 15 by green crosses. They allow us to impose restrictions to the size and to the orientation of the bar. These masers are associated with massive star formation regions and molecular clouds. Ob- servations of external barred galaxies tells us that there is no star formation inside bars (James & Percival 2018). So, the abrupt cut in the space density of masers at the nearest extremity of the bar tells us that the size of the bar drawn in the figure is approx- imately correct, confirming its length of the order of 3 ...
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... other two regions are plotted by the levels with light- grey tones in Figure 4. In the intermediate zone, the loci of the azimuthal minima of the potential H 1 (R, ϕ) (blue curves), which, outside the zone of influence of the bar, correspond to the loci of the main spiral arms in our model, are strongly perturbed; this is noted from the comparison with the unperturbed spiral arms shown in Figure 1. Two arms connected with the extremals of the bar, where the bar's force function is strongest, suffer significant deformation, while the other two arms are partially vanishing. We estimate that the intermediate zone extends up to ∼4.3 kpc until its effects disappear. It is interesting to note that the upper boundary of the intermediate zone matches closely the position of the "extended bar", which some authors observe at Galac- tocentric distances of 4-4.5 kpc (e.g. López-Corredoira et al. ...
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... Figure 11 shows that, for M bar = 10 9 M , the zone of the bar's influence never approaches the domain of the spiral corotation, where the Sun is evolving. Thus, it is expected that, for the adopted bar's mass, the motion of the Sun and the location of the L 4 -center on the X-Y plane is only slightly affected by increasing f bar ...
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... order to outline the inner region, which we will refer to as the zone of influence of the bar, we plot the energy levels in hard black tones in Figure 4. The zone of influence is extended up to ∼3.5 kpc along the main axis of the bar, that is, beyond the physical extension of the bar with R bar = 2.9 kpc (see Table 1). The geometry of the zone of influence also differs from that of an elongated ellipsoid which shapes the bar (see Figure 1); it is rather alike an observable box-shaped ...
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... model of the bar predicts the existence of four regions around it of complicated geometry: the zig-zags in the grooves which represent the minima of gravitational potential, at the ex- tremities of the bar, and the short sectors of spiral arms, at 90 • Article number, page 12 of 17 from the main axis. We numbered these regions from 1 to 4, in Figure 15. The Galactic longitudes of these regions can be easily determined graphically by tracing lines joining them to the posi- tion of the Sun and measuring their angle with respect to the Y axis as shown in the Figure; they are, respectively, ...
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... test the effects of the bar's size on the Sun's motion, which is representative of the dynamical stability of objects from the Local Arm. For this, we construct the dynamical map on the parametric plane M bar -R bar shown in Figure 12. Varying the bar's mass in the range between 10 8 M and 1.5×10 10 M and the bar's radius between 1 kpc to 6 kpc, we analyse the dynamical stabil- ity of the Sun, with coordinates X = 0 and Y = 8.0 kpc and the velocities p R = −11 km s −1 and V θ = 242.24 km s −1 . The rest of the parameters is taken from Table 1. The perturbations to stellar motion due to the bar are weak in the light-tone domain on the dynamical map in Figure 12; they increase in the darker zones and provoke strong instabili- ties in the red hatched region. (The fine effects in the solar mo- tion shown by slight variations of gray tones in the domain of stable orbits are not analysed here.) The analysis of the map shows the effects of the mass of the bar and of its radius on the stability of the solar orbit: the increasing mass reduces the stability of the stellar motion inside the corotation zone, while the decreasing radius enhances this stability. Both parameters are saturated: the bar's mass at ∼ 10 9 M and the bar's radius at ∼1.7 kpc. For the adopted value R bar = 2.9 kpc, the current mo- tion of the Sun and objects from the Local Arm remains stable up to M bar ≈ 5 × 10 9 M ...
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... test the effects of the bar's size on the Sun's motion, which is representative of the dynamical stability of objects from the Local Arm. For this, we construct the dynamical map on the parametric plane M bar -R bar shown in Figure 12. Varying the bar's mass in the range between 10 8 M and 1.5×10 10 M and the bar's radius between 1 kpc to 6 kpc, we analyse the dynamical stabil- ity of the Sun, with coordinates X = 0 and Y = 8.0 kpc and the velocities p R = −11 km s −1 and V θ = 242.24 km s −1 . The rest of the parameters is taken from Table 1. The perturbations to stellar motion due to the bar are weak in the light-tone domain on the dynamical map in Figure 12; they increase in the darker zones and provoke strong instabili- ties in the red hatched region. (The fine effects in the solar mo- tion shown by slight variations of gray tones in the domain of stable orbits are not analysed here.) The analysis of the map shows the effects of the mass of the bar and of its radius on the stability of the solar orbit: the increasing mass reduces the stability of the stellar motion inside the corotation zone, while the decreasing radius enhances this stability. Both parameters are saturated: the bar's mass at ∼ 10 9 M and the bar's radius at ∼1.7 kpc. For the adopted value R bar = 2.9 kpc, the current mo- tion of the Sun and objects from the Local Arm remains stable up to M bar ≈ 5 × 10 9 M ...
Citations
... A a espessura, na direção perpendicular ao plano, é da ordem de 1,5 kpc Em praticamente todas as representações do disco galáctico que incluem a barra, esta é representada por um oval, sem maiores detalhes sobre a forma real. Apresentamos na Figura 14 o desenho da barra segundo Michtchenko et al (2018) [11], no qual o formato externo é parecido com um retângulo. ...
... A a espessura, na direção perpendicular ao plano, é da ordem de 1,5 kpc Em praticamente todas as representações do disco galáctico que incluem a barra, esta é representada por um oval, sem maiores detalhes sobre a forma real. Apresentamos na Figura 14 o desenho da barra segundo Michtchenko et al (2018) [11], no qual o formato externo é parecido com um retângulo. ...
... á um consenso de que a barra é constituída de estrelas velhas e pouco luminosas, com baixa metalicidade, precisamente como acontece no bojo, e em contraste com o disco, no qual as estrelas mais jovens são abundantes Para comparação na Figura 15 apresentamos o modelo da barra por Bland Hawthorn e Gherard (2016). A imagem do centro, com a barra vista de cima tem semelhança com a Figura 14 (de Mitchenko et al [10], veja também [11]) se consideramos a contorno colorido em verde. A imagem de cima mostra de forma clara que a barra está na continuidade do bojo. ...
Apresentamos a Via Láctea, nossa Galáxia, primeiro com uma introdução de como a vemos no céu, e a seguir com a história dos primeiros passos de descobertas, que levaram a conclusão de que a Galáxia não é o próprio Universo, mas apenas uma entre muitas galáxias.Descrevemos seus componentes principais (disco galáctico, braços espirais, bojo, barra), mencionando suas dimensões e formas geométricas. Focalizamos as órbitas das estrelas, responsáveis pela manutenção da forma dos componentes. Explicamos as órbitas, a partir das forças gravitacionais que atuam sobre as estrelas
... The influence of the OLR resonances on the distribution of particles is also illustrated in the (R, V φ )-diagram of Figure 7, which shows the appearance of noticeable diagonal ridges seen on the figure. Similar ridges, confirmed observationally [6], are connected to the resonant orbital structure in the barred potential that forms regions in the phase space with stable and unstable orbits, and hence the regions with overdensities and gaps [5,65,66]. To demonstrate the influence of resonances occurring in the plane of the disk on the disk's vertical structure, we follow Trick et al. [22] and describe the dynamics of the disk in (L Z , < |V Z | >) and (L Z , < |Z| >) space. ...
Detailed analysis of kinematics of the Milky Way disk in the solar neighborhood based on the GAIA DR3 catalog reveals the existence of peculiarities in the stellar velocity distribution perpendicular to the galactic plane. We study the influence of resonances—the outer Lindblad resonance and the outer vertical Lindblad resonance—of a rotating bar with stellar oscillations perpendicular to the plane of the disk, and their role in shaping the spatial and the velocity distributions of stars. We find that the Z and VZ distributions of stars with respect to LZ are affected by the outer Lindblad resonance. The existence of bar resonance with stellar oscillations perpendicular to the plane of the disk is demonstrated for a long (large semi-axis 5 kpc) and fast rotating bar with Ωb=60.0kms−1kpc−1. We show also that, in the model with the long and fast rotating bar, some stars in the 2:1 OLR region deviate far from their original places, entering the bar region. A combination of resonance excitation of stellar motions at the 2:1 OLR region together with strong interaction of the stars with the bar potential leads to the formation of the group of ‘escapees’, i.e., stars that deviate in R and Z—directions at large distances from the resonance region. Simulations, however, do not demonstrate any noticeable effect on VZ-distribution of stars in the solar neighborhood.
... They are also possible origins for the Hercules stream and the other velocity-space substructures (Monari et al. 2017c ;Hattori et al. 2019 ;Monari et al. 2019a ;Moreno et al. 2021 ). Spiral arm resonances or their combination with bar resonances also form velocity-space substructures Michtchenko et al. 2018aMichtchenko et al. , 2019Hattori et al. 2019 ;Barros et al. 2020 ). ...
The velocity-space distribution of the solar neighbourhood stars shows complex substructures. Most of the previous studies use static potentials to investigate their origins. Instead we use a self-consistent N-body model of the Milky Way, whose potential is asymmetric and evolves with time. In this paper, we quantitatively evaluate the similarities of the velocity-space distributions in the N-body model and that of the solar neighbourhood, using Kullback-Leibler divergence (KLD). The KLD analysis shows the time evolution and spatial variation of the velocity-space distribution. The KLD fluctuates with time, which indicates the velocity-space distribution at a fixed position is not always similar to that of the solar neighbourhood. Some positions show velocity-space distributions with small KLDs (high similarities) more frequently than others. One of them locates at $(R,\phi )=(8.2\,\,{\rm{kpc}}, 30^\circ )$, where R and φ are the distance from the galactic centre and the angle with respect to the bar’s major axis, respectively. The detection frequency is higher in the inter-arm regions than in the arm regions. In the velocity maps with small KLDs, we identify the velocity-space substructures, which consist of particles trapped in bar resonances. The bar resonances have significant impact on the stellar velocity-space distribution even though the galactic potential is not static.
... coherent kinematic structures in the local xy space) are either dynamical footprints of the Galactic spiral arms and bar (e.g. Antoja et al. 2008 ;Michtchenko et al. 2018 ), or dissolving open clusters and associations (e.g. Oh & Evans 2020 ;Gagn é et al. 2021 ). ...
Gravitational acceleration fields can be deduced from the collisionless Boltzmann equation, once the distribution function is known. This can be constructed via the method of normalizing flows from datasets of the positions and velocities of stars. Here, we consider application of this technique to the solar neighbourhood. We construct mock data from a linear superposition of multiple ‘quasi-isothermal’ distribution functions, representing stellar populations in the equilibrium Milky Way disc. We show that given a mock dataset comprising a million stars within 1 kpc of the Sun, the underlying acceleration field can be measured with excellent, sub-percent level accuracy, even in the face of realistic errors and missing line-of-sight velocities. The effects of disequilibrium can lead to bias in the inferred acceleration field. This can be diagnosed by the presence of a phase space spiral, which can be extracted simply and cleanly from the learned distribution function. We carry out a comparison with two other popular methods of finding the local acceleration field (Jeans analysis and 1D distribution function fitting). We show our method most accurately measures accelerations from a given mock dataset, particularly in the presence of disequilibria.
... For example, moving groups (i.e. coherent kinematic structures in the local xy space) are either dynamical footprints of the Galactic spiral arms and bar (e.g., Antoja et al. 2008;Michtchenko et al. 2018), or dissolving open clusters and associations (e.g., Oh & Evans 2020;Gagné et al. 2021). These could provide additional contributions to the systematic bias in our estimation of the Galactic acceleration field, but a full accounting is beyond the scope of the present work. ...
Gravitational acceleration fields can be deduced from the collisionless Boltzmann equation, once the distribution function is known. This can be constructed via the method of normalizing flows from datasets of the positions and velocities of stars. Here, we consider application of this technique to the solar neighbourhood. We construct mock data from a linear superposition of multiple `quasi-isothermal' distribution functions, representing stellar populations in the equilibrium Milky Way disc. We show that given a mock dataset comprising a million stars within 1 kpc of the Sun, the underlying acceleration field can be measured with excellent, sub-percent level accuracy, even in the face of realistic errors and missing line-of-sight velocities. The effects of disequilibrium can lead to bias in the inferred acceleration field. This can be diagnosed by the presence of a phase space spiral, which can be extracted simply and cleanly from the learned distribution function. We carry out a comparison with two other popular methods of finding the local acceleration field (Jeans analysis and 1D distribution function fitting). We show our method most accurately measures accelerations from a given mock dataset, particularly in the presence of disequilibria.
... They are also possible origins for the Hercules stream and the other velocity-space substructures (Monari et al. 2017c;Hattori et al. 2019;Monari et al. 2019a;Moreno et al. 2021). Spiral arm resonances or their combination with bar resonances also form velocity-space substructures Hattori et al. 2019;Michtchenko et al. 2018aMichtchenko et al. , 2019Barros et al. 2020). ...
The velocity-space distribution of the solar neighbourhood stars shows complex substructures. Most of the previous studies use static potentials to investigate their origins. Instead we use a self-consistent $N$-body model of the Milky Way, whose potential is asymmetric and evolves with time. In this paper, we quantitatively evaluate the similarities of the velocity-space distributions in the $N$-body model and that of the solar neighbourhood, using Kullback-Leibler divergence (KLD). The KLD analysis shows the time evolution and spatial variation of the velocity-space distribution. The KLD fluctuates with time, which indicates the velocity-space distribution at a fixed position is not always similar to that of the solar neighbourhood. Some positions show velocity-space distributions with small KLDs (high similarities) more frequently than others. One of them locates at $(R,\phi)=(8.2\;\mathrm{kpc}, 30^{\circ})$, where $R$ and $\phi$ are the distance from the galactic centre and the angle with respect to the bar's major axis, respectively. The detection frequency is higher in the inter-arm regions than in the arm regions. In the velocity maps with small KLDs, we identify the velocity-space substructures, which consist of particles trapped in bar resonances. The bar resonances have significant impact on the stellar velocity-space distribution even though the galactic potential is not static.
... This has been modelled extensively assuming it to be the Outer Lindblad resonance (OLR) of a 50 < Ω b [km s −1 kpc −1 ] < 60 bar (Dehnen 2000;Fux 2001;Minchev et al. 2010;Antoja et al. 2014;Monari et al. 2017). More recently, Hercules has been modelled as a corotation (CR) resonance (Pérez-Villegas et al. 2017;Monari et al. 2019b;D'Onghia & L. Aguerri 2020;, the 4:1 OLR of a longslow bar , a combination of both CR & OLR (Asano et al. 2020;Kawata et al. 2020), or a combination of bar and spiral arm resonances (Quillen 2003;De Simone et al. 2004;Chakrabarty 2007; Antoja et al. 2009;Michtchenko et al. 2018;Hat-tori et al. 2019). However Hercules is just one of a number of local kinematic observations yet to be consistently explained. ...
We compare distance resolved, absolute proper motions in the Milky Way bar/bulge region to a grid of made-to-measure dynamical models with well defined pattern speeds. The data are obtained by combining the relative VVV Infrared Astrometric Catalog v1 proper motions with the Gaia DR2 absolute reference frame. We undertake a comprehensive analysis of the various errors in our comparison, from both the data and the models, and allow for additional, unknown, contributions by using an outlier-tolerant likelihood function to evaluate the best fitting model. We quantify systematic effects such as the region of data included in the comparison, with or without possible overlap from spiral arms, and the choice of synthetic luminosity function and bar angle used to predict the data from the models. Resulting variations in the best-fit parameters are included in the final error budget. We measure the bar pattern speed to be Omega_b=35.4+-0.9 km/s/kpc and the azimuthal solar velocity to be V_phi_sun= 251.4+-1.7 km/s. These values, when combined with recent measurements of the Galactic rotation curve, yield the distance of corotation, 6.3 < R_(CR) [kpc] < 6.8, the outer Lindblad resonance (OLR), 10.5 < R_(OLR) [kpc] < 11.5, and the higher order, m=4, OLR, 8.5 < R_(OLR_4) [kpc] < 9.0. The measured low pattern speed provides strong evidence for the "long-slow" bar scenario.
... kpc and with an axial ratio of about 1:0.4 is another peculiarity of the Milky Way galaxy [77]. The major semi-axis of the bar deviates by approximately 23 • from the direction to the Galactic center [81]. A successful dynamical model of the Milky Way must agree with the available observational data and reproduce the observed parameters of the bar. ...
... As one can see, the rotation curve of the disk along the semi-minor-axis of the bar has a local maximum inside 1 kpc from the center of galaxy, similar to the observed kinematical properties of the Milky Way disk within the central kiloparsec (see Figure 1). The estimates of the angular velocity and mass of the Milky Way bar [81] provide values of Ω bar < 50 km s −1 kpc −1 and M bar 2 × 10 9 M , respectively. ...
... The authors [122] made a fundamental conclusion, that stabilization of the bar mode requires the value of the parameter µ within (1-2). Criteria of this kind have been tested using better N-body models and they show that the exact value of µ depends on a number of parameters of the galactic subsystems that determine the properties of the bulge, the gaseous disk, the radial profile of the dark matter, etc. [22,27,81,89,123]. Similarly, in the models with a moderate ratio of masses of the halo and of the disk µ 1.6, but with a massive and compact bulge b/R opt 0.014, the bar is not reproduced in the numerical simulations due to scattering of the orbits of the collisionless particles in the central regions where the gravitational potential has a large gradient. ...
Using recent observational data, we construct a set of multi-component equilibrium models of the disk of a Milky Way-like galaxy. The disk dynamics are studied using collisionless-gaseous numerical simulations, based on the joined integration of the equations of motion for the collision-less particles using direct integration of gravitational interaction and the gaseous SPH-particles. We find that after approximately one Gyr, a prominent central bar is formed having a semi-axis length of about three kpc, together with a multi-armed spiral pattern represented by a superposition of $m=$ 2-, 3-, and 4-armed spirals. The spiral structure and the bar exist for at least 3 Gyr in our simulations. The existence of the Milky Way bar imposes limitations on the density distributions in the subsystems of the Milky Way galaxy. We find that a bar does not form if the radial scale length of the density distribution in the disk exceeds 2.6 kpc. As expected, the bar formation is also suppressed by a compact massive stellar bulge. We also demonstrate that the maximum value in the rotation curve of the disk of the Milky Way galaxy, as found in its central regions, is explained by non-circular motion due to the presence of a bar and its orientation relative to an observer.
... kpc and with an axial ratio of about 1:0.4 is another peculiarity of the Milky Way galaxy [77]. The major semi-axis of the bar deviates by approximately 23 • from the direction to the Galactic center [81]. A successful dynamical model of the Milky Way must agree with the available observational data and reproduce the observed parameters of the bar. ...
... The estimates of the angular velocity and mass of the Milky Way bar [81] provide values of Ω bar < 50 km s −1 kpc −1 and M bar 2 × 10 9 M , respectively. ...
... The authors [122] made a fundamental conclusion, that stabilization of the bar mode requires the value of the parameter µ within (1)(2). Criteria of this kind have been tested using better N-body models and they show that the exact value of µ depends on a number of parameters of the galactic subsystems that determine the properties of the bulge, the gaseous disk, the radial profile of the dark matter, etc. [22,27,81,89,123]. Let us estimate the influence of the relative mass of the halo on the velocity dispersion of stars that are close to the boundary of stability of the system. ...
Using recent observational data, we construct a set of multi-component equilibrium models of the disk of a Milky Way-like galaxy. The disk dynamics are studied using collisionless-gaseous numerical simulations, based on the joined integration of the equations of motion for the collision-less particles using direct integration of gravitational interaction and the gaseous SPH-particles. We find that after approximately one Gyr, a prominent central bar is formed having a semi-axis length of about three kpc, together with a multi-armed spiral pattern represented by a superposition of m= 2-, 3-, and 4-armed spirals. The spiral structure and the bar exist for at least 3 Gyr in our simulations. The existence of the Milky Way bar imposes limitations on the density distributions in the subsystems of the Milky Way galaxy. We find that a bar does not form if the radial scale length of the density distribution in the disk exceeds 2.6 kpc. As expected, the bar formation is also suppressed by a compact massive stellar bulge. We also demonstrate that the maximum value in the rotation curve of the disk of the Milky Way galaxy, as found in its central regions, is explained by non-circular motion due to the presence of a bar and its orientation relative to an observer.
... The spiral parameters values presented above have been widely tested and used in the works by our group (Barros et al., 2016(Barros et al., , 2020Lépine et al., 2017;Michtchenko et al., 2017Michtchenko et al., , 2018a, as well as found to give a self-consistent picture of the spiral structure in its original form by Junqueira et al. (2013). For instance, the ζ 0 -value, which gives the spiral-arm strength, is adjusted to guarantee the self-consistency of the arm shape . ...
... In the present work, we adopt the value of p = 28.5 km s −1 kpc −1 , as used by Lépine et al. (2017), and Michtchenko et al. (2018a) in a series of papers by our group. This value is in close agreement with the recent determination by Dias et al. (2019) using the Gaia DR2 data. ...
... By studying the dynamical effects due to the coupling between the perturbations from the bar and the spiral arms on the kinematics of the SN, Michtchenko et al. (2018a) constrained the physical parameters of an elongated bar model (size, mass, flattening, orientation) and regarding the bar pattern speed, their model favored b < 50 km s −1 kpc −1 for an allowed bar mass of ∼ 2 × 10 9 M ⊙ . These values were obtained by taking the requirement for the stability of the Local Arm structure that is a consequence of the dynamical stability of the corotation zone of the spiral pattern (see section 5.1), as well as the stability of the bar itself. ...
This article discusses the effects of the spiral-arm corotation on the stellar dynamics in the Solar Neighborhood (SN). All our results presented here rely on: (1) observational evidence that the Sun lies near the corotation circle, where stars rotate with the same angular velocity as the spiral-arm pattern; the corotation circle establishes domains of the corotation resonance (CR) in the Galactic disk; (2) dynamical constraints that put the spiral-arm potential as the dominant perturbation in the SN, comparing with the effects of the central bar in the SN; (3) a long-lived nature of the spiral structure, promoting a state of dynamical relaxing and phase-mixing of the stellar orbits in response to the spiral perturbation. With an analytical model for the Galactic potential, composed of an axisymmetric background deduced from the observed rotation curve, and perturbed by a four-armed spiral pattern, numerical simulations of stellar orbits are performed to delineate the domains of regular and chaotic motions shaped by the resonances. Such studies show that stars can be trapped inside the stable zones of the spiral CR, and this orbital trapping mechanism could explain the dynamical origin of the Local arm of the Milky Way (MW). The spiral CR and the near high-order epicyclic resonances influence the velocity distribution in the SN, creating the observable structures such as moving groups and their radially extended counterpart known as diagonal ridges. The Sun and most of the SN stars evolve inside a stable zone of the spiral CR, never crossing the main spiral-arm structure, but oscillating in the region between the Sagittarius-Carina and Perseus arms. This orbital behavior of the Sun brings insights to our understanding of questions concerning the solar system evolution, the Earth environment changes, and the preservation of life on Earth.
