A NEW MECHANISM FOR RADIAL MIGRATION IN GALACTIC DISKS: SPIRAL–BAR RESONANCE OVERLAP

The metallicity of the interstellar medium (ISM) in a galaxy is expected to increase with time due to the progressive enrichment through stellar feedback. Consequently, one would expect that younger stars formed at the same galactic radius would have higher metallicities. In addition, the ISM metallicity has been well established to decrease with increasing galactic radius, r, due to a more efficient star formation and enrichment of the ISM in the central regions of galaxies (Wilson & Rood 1994; Daflon & Cunha 2004). This in turn would result in coeval stars being progressively more metal-poor as r increases.

However, the metallicities of stars in the solar neighborhood (SN) have been observed to lack the expected correlation with age (Edvardsson et al. 1993; Haywood 2008). Re-examining the age–metallicity distribution of the Geneva–Copenhagen survey (Nordström et al. 2004), Haywood (2008) found that the data are only consistent with the age–metallicity relation (AMR) for stars younger than 3 Gyr. For other samples the width of the metallicity distribution was found to be larger than what could be accounted for by measurement errors and/or inhomogeneous ISM. Because of the age–velocity relation (AVR), where older stellar samples are observed to have higher velocity dispersion than younger ones (e.g., Holmberg et al. 2009), stars from the inner and outer Milky Way (MW) disk can enter the SN and thus blur the expected AMR. However, this can only account for about 50% of the observed scatter (Nordström et al. 2004).

A possible explanation for this discrepancy is the radial migration of stars from their birth guiding radii (Haywood 2008; Schönrich & Binney 2009). Indeed, there is now considerable evidence that the non-axisymmetry of the Galactic potential can cause significant perturbations in the motion of both stars and gas: observational evidence for this comes from (1) the observed non-circular motions of gas flows in the inner MW (Bissantz et al. 2003), (2) the large non-axisymmetric motions of star-forming regions (Xu et al. 2006), and (3) the moving groups in the SN (Dehnen 1998) containing stars of very different ages (Famaey et al. 2005; Antoja et al. 2008), suggesting that their clumping in velocity space is most likely not due to irregular star formation, but rather to dynamical perturbations from the bar (Dehnen 2000; Minchev et al. 2010a) and/or the spirals (e.g., Quillen & Minchev 2005; Antoja et al. 2009), or an orbiting satellite galaxy (Quillen et al. 2009; Minchev et al. 2009).

A numerically well-studied effect of this non-axisymmetry of the Galactic potential is the radial migration of stars due to resonant scattering by transient spiral structure (SS; Sellwood & Binney 2002, hereafter SB02; Roškar et al. 2008), which could explain the aforementioned absence of AMR in the SN. N-body simulations based on cosmological initial conditions (e.g., Heller et al. 2007) would a priori be the most natural framework in which to study radial mixing in galactic disks. However, these simulations cannot at present produce large enough thin galactic disks due to the transfer of angular momentum from the disk to the live dark matter halo. The resulting galaxies are typically thick and present large stellar velocity dispersions, so that they are closer to S0 galaxies than to MW-like ones (Sánchez-Blázquez et al. 2009). On the other hand, Roškar et al. (2008) used an N-body hydrodynamical model in which the initial conditions were designed to mimic the stage following a last major merger when the thin disk formation is supposed to start. Both SB02 and Roškar et al. (2008) compare the effects of radial migration in the MW disk, for which the existence of a bar is firmly established, to a simulated galaxy lacking a central bar. Unless the effect of the bar is unimportant to the mixing process, their results would only poorly approximate the radial migration in our Galaxy. In this paper, we check whether the effect of galactic bars can indeed be neglected in the context of radial migration in galactic disks, as assumed in a number of previous studies. For this purpose, we study the combined effect of SS and a central bar (and more generally of multiple patterns) on the radial mixing of stars in a stellar disk.

There is observational evidence for multiple patterns in external galaxies, such as asymmetries in the SS (e.g., Henry et al. 2003; Meidt et al. 2008). By expanding galaxy images in Fourier components, Elmegreen et al. (1992) noted that many galaxies exhibit hidden three-armed components and suggested that multiple spiral density waves can propagate simultaneously in galaxy disks. In addition, by means of the recently developed (Merrifield et al. 2006) radial Tremaine–Weinberg method, Meidt et al. (2009) analyzed the high-quality H i and CO data cubes available for four spiral galaxies. The authors found direct evidence for the presence of resonant coupling of multiple patterns, including spiral–spiral and spiral–bar components.

The results of N-body simulations provide additional motivation for studying the dynamical effects of multiple patterns. Sellwood (1985) and Sellwood & Sparke (1988) presented N-body experiments in which a bar coexists with a spiral pattern moving at a much lower angular velocity. Tagger et al. (1987) and Sygnet et al. (1988) explained this as a nonlinear mode coupling between a bar and SS. Sellwood's simulations, and the theoretical explanation of Tagger et al. (1987) and Sygnet et al. (1988), were later confirmed by Masset & Tagger (1997) and Rautiainen & Salo (1999).

How do multiple patterns affect the dynamics of galactic disks? Quillen (2003) considered the dynamics of stars that are affected by perturbations from both SS and a central bar by constructing a one-dimensional Hamiltonian model for the strongest resonances in the epicyclic action-angle variables. Quillen pointed out that when two perturbers with different pattern speeds are present in the disk, the stellar dynamics can be stochastic, particularly near resonances associated with one of the patterns. Similar findings were presented more recently by Jalali (2008). All these results are not surprising since it has already been shown by Chirikov (1979) that in the case of resonance overlap the last Kolmogorov–Arnold–Moser surface between the two resonances is destroyed, resulting in stochastic behavior. We therefore expect that resonance overlap could give rise to both velocity dispersion increase and radial migration.

In this paper, we study the combined effect of a central bar and SS on the dynamics of a galactic disk. We concentrate on a steady-state SS in order to assess the pure effect of nonlinear coupling between the two perturbers, noting that while there is strong circumstantial evidence that SS is mostly non-stationary in unbarred galaxies, the situation is less clear in barred ones (Binney & Tremaine 2008). In any case, the following model can also serve as a proxy for studying the effect of transients, and is complementary to the transient spiral mechanism developed by SB02.

Galactic disks rotate differentially with nearly flat rotation curves, i.e., constant circular velocity as a function of galactic radius. In contrast, density waves, such as a central bar or SS, rotate as solid bodies. Therefore, stars at different radii would experience different forcing due to the non-axisymmetric structure. Of particular interest to us are locations in the disk where the stars are in resonance with the perturber. The corotation resonance (CR), where stars move with the pattern, occur when the angular rotation rate of stars equals that of the perturber. The Lindblad resonances (LRs) occur when the frequency at which a star feels the force due to a perturber coincides with the star's epicyclic frequency, κ. As one moves inward or outward from the corotation circle, the relative frequency at which a star encounters the perturber increases. There are two values of r for which this frequency is the same as the radial epicyclic frequency. This is where the inner and outer Lindblad resonances (ILR and OLR, respectively) are located. Quantitatively, LRs occur when Ω_b,s = Ω ± κ/m, where m is the multiplicity of the pattern. The negative sign corresponds to the ILR and the positive to the OLR. While Bertil Lindblad defined these for the case of an m = 2 pattern (thus strictly speaking the ILR/OLR are the 2:1 resonances), for an m = 4 pattern the ILR/OLR must be the 4:1 resonances.

Since second-order resonances, i.e., 4:1 for a two-armed spiral or bar, or 2:1 for a four-armed spiral, can also be quite important (as will be shown later), we need to have a convenient way to refer to them. It is somewhat confusing and unclear how the 4:1 resonances are referred to in the literature. The inner 4:1 resonance for an m = 2 pattern is known as the ultra-harmonic resonance (UHR). Some also describe the inner and outer 4:1 resonances as the IUHR and the OUHR, others as the inner and outer m = 4 resonance. If the pattern multiplicity is m = 4, then these become the ILR and OLR. To our knowledge, there is no terminology for the 2:1 resonances of an m = 4 pattern. To simplify things, in this paper we generalize the standard notation of Lindblad resonances to allow for the existence of both 2:1 and 4:1 resonances, regardless of the multiplicity of the pattern. In other words, we will refer to the 2:1 ILR/OLR and the 4:1 ILR/OLR for both two-armed (or bar) and four-armed SS. Naturally, other resonances can also be described in this manner, i.e., 3:1, 5:1, 6:1 ILR/OLR, etc.

Figure 1 illustrates the relationship between the pattern angular velocity and the radii at which resonances occur for a flat rotation curve and nearly circular orbits. Note that for two (or more) non-axisymmetric patterns (such as bar + SS or SS + SS) moving at different angular velocities there will always exist regions in the disk where resonances overlap. As we will see later, this turns out to be crucial for the dynamics of barred galaxies.

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We follow the motion of test particles in an initially axisymmetric, two-dimensional stellar disk with a flat rotation curve as described in detail in Minchev & Quillen (2007), in which we grow non-axisymmetric perturbations as described below. Both the initial disk stellar density, Σ(r), and the radial velocity dispersion, σ_r(r), decrease exponentially with radius, where we initially start with σ_r(r_d) = 5 km s⁻¹ at the disk scale length, r_d = 3 kpc for an MW-type galaxy. To simulate the effect of a central bar and spiral density waves we impose perturbations on the initially axisymmetric disk as done by Minchev et al. (2007) and Minchev & Quillen (2007), respectively. Both bar and spiral perturbations are grown simultaneously in 0.4 Gyr by increasing their amplitudes from zero to the maximum values. To ensure a smooth transition to the perturbed state we use the function defined in Equation (4) from Dehnen (2000). To be able to relate parameters to the MW, we work in units of the solar distance and circular velocity with corresponding values r₀ = 8 kpc and v_c = 240 km s⁻¹; this gives Ω₀ = 30 km s⁻¹ kpc⁻¹.

While in Minchev & Quillen (2007) and Minchev et al. (2007) we required the integration of a large number of particles in order to be able to resolve an "SN," here we are interested in the global behavior of a galactic disk. Therefore, we have found it sufficient to integrate 10⁵ particles for each simulation, distributed between an inner and outer radii, r_in = 2 and r_out = 15 kpc. We have verified that increasing the resolution to 10⁷ particles does not change our results. We divide each sample into 50 radial bins and calculate the azimuthally averaged changes in angular momentum in the disk in order to estimate the induced radial migration. The time evolution of each simulation is followed for 3 Gyr. We distinguish between the SS and the bar by indicating the OLR as OLR_s and OLR_b, respectively, and similarly for the case of the ILR.

3.1. Perturbation from a Central Bar

The bar strength, _b, for the case of the MW is defined as the ratio of the forces due to the bar's quadrupole and the axisymmetric background at the Sun's radius, r₀, on the bar's major axis, _b = 3(A_b/v²_c)(r_b/r₀)³ (Dehnen 2000), where v_c is the asymptotic circular velocity. The bar potential is then given by

$$\begin{equation} \Phi _{\rm b} = A_{\rm b}(\epsilon _{\rm b}) \cos [2(\phi -\Omega _{\rm b} t)]\times \left\lbrace \begin{array}{@{}cclcr@{}} \left(r_{\rm b}\over r\right)^3,& r \ \ge \ r_{\rm b} \\ \ms 2-\left(r\over r_{\rm b}\right)^3,& r \ \le \ r_{\rm b}, \end{array} \right. \end{equation} \tag{ 1 }$$

where $r_b=0.8r_{\rm{cr}}$ is the bar half-length in terms of its CR, $r_{\rm{cr}}$; in physical units, r_b = 3.44 kpc and $r_{\rm{cr}}=4.3$ kpc.

Previous work has used as a measure of bar strength the parameter Q_T (Combes & Sanders 1981). This is the ratio of the maximum tangential force to the azimuthally averaged radial force at a given radius. From Equation (1) this definition yields Q_T = 2A_b/v²_c. We examine bar amplitudes in the range 0.1 < Q_T < 0.4 as expected from observations of various galaxies and from N-body simulations (Combes & Sanders 1981). This corresponds to 0.013 < |_b| < 0.05 in our units.

The bar pattern speed is chosen to be Ω_b = 55.5 km s⁻¹ kpc⁻¹, which for the case of our Galaxy can be written as Ω_b = 1.85Ω₀. This value is consistent with observations of external galaxies where it is found that bars are dynamically fast, i.e., $r_{\rm{cr}}/r_b = 1.2\pm 0.2$ (Aguerri et al. 2003), and is also in accordance with recent bar pattern speed determinations for the MW (Minchev et al. 2010a, 2007; Bissantz et al. 2003; Dehnen 2000).

3.2. Perturbation from Spiral Structure

The spiral potential is given by

$$\begin{equation} \Phi _s(r,\phi,t) = \epsilon _s \cos \left[\alpha \ln {r\over r_0}-m(\phi -\Omega _s t)\right], \end{equation} \tag{ 2 }$$

where _s is the SS strength, related to the amplitude of the mass surface density of spirals, Σ_s, as

$$\begin{equation} \epsilon _s \approx - 2 \pi G \Sigma _s r_0 / \big(\alpha v_c^2\big), \end{equation} \tag{ 3 }$$

as shown in Binney & Tremaine (2008). The parameter α is related to the pitch angle of the spirals, p, by $\alpha = m \cot (p)$. The azimuthal wavenumber m is an integer corresponding to the number of arms. We consider both two-armed and four-armed SS with α = −4 and −8, respectively, where the negative sign corresponds to trailing spirals. Elmegreen (1998) found that grand-design spirals have arm–interarm contrasts of 1.5–6, corresponding to a fractional amplitude of 0.2 < Σ_s/Σ < 0.7, which is in agreement with Rix & Zaritsky (1995) who estimated 0.15 < (Σ_s/Σ) < 0.6. Since we would like to examine this range of spiral amplitudes, we need to relate the relative overdensity to the relative potential.

For a maximum exponential disk, the peak circular speed in the disk has the value $v_c \simeq 0.622 \sqrt{GM_d / r_d}$ at r ≃ 2.15r_d, where M_d is the disk mass inclosed by r_d. The surface density of the disk at radius r is $\Sigma (r) = ({M_d / 2\pi r_d^2}) e^{-r/r_d}$. Eliminating M_d from these expressions, we find at r₀, $v_c^2 \simeq 0.39 \times 2\pi G r_d\Sigma _0 e^{r_0/r_d}$. Substituting this expression for v_c in Equation (3), the relation between the relative potential and the relative overdensity becomes

$$\begin{equation} \epsilon _s \approx - {\Sigma _s\over \Sigma _0} {r_0 \over r_d} {e^{-r_0\over r_d} \over 0.39 \alpha }. \end{equation} \tag{ 4 }$$

To correct for finite disk thickness we need to decrease the right-hand side of this equation by the factor $e^{-kz_0}$ (e.g., Vandervoort 1970), where k is the wave vector given by k ≈ α/r₀. For an MW-type galaxy we can adopt a disk scale height of z₀ ∼ 300 pc, resulting in $e^{-kz_0}\approx 0.86$ and 0.74 for a two- and four-armed SS, respectively. Using Equation (4) and accounting for finite disk thickness we estimate that the observed spiral arm amplitudes quoted above would result in potential perturbation in the range 0.007 < |_s| < 0.031 for a four-armed SS and 0.015 < |_s| < 0.072 for a two-armed SS.

We now estimate the changes in angular momentum, ΔL, in a stellar disk. To do this, we divide the disk into 50 radial bins and compute ΔL = L₁ − L₀, where L₀ and L₁ are the mean values of the initial (at t = 0) and final (at some later time, t₁) angular momenta of particles in each bin.

In Figure 2, we show ΔL as a function of L₀ for a simulation with a single perturber. Both axes are in units of the circular velocity, v_c; L₀ is thus equivalent to the galactocentric distance, r in kpc. The bar and SS amplitudes are indicated in each panel. The first two panels from left to right present simulations with a bar of an intermediate strength (Q_T = 0.25) and a strong one (Q_T = 0.4), respectively. The third and fourth panels show a four-armed SS with relative overdensity Σ_s/Σ₀ = 0.2, 0.35. The rightmost panel presents a simulation with a two-armed SS with an amplitude giving rise to Σ_s/Σ₀ = 0.25. The dotted lines show the corotation radii. The 2:1 and 4:1 LRs are indicated by the solid and dashed vertical lines, respectively (bar = red, spiral = blue). Note that, depending on the pattern speed, certain resonances might not be present in the disk. The pattern speeds are given in units of the angular velocity at r₀ = 8 kpc, Ω₀ = 30 km s⁻¹ kpc⁻¹.

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Significant angular momentum changes in Figure 2 occur only near the CR of each perturber (dotted lines). No further increase in |ΔL| is apparent with time once the perturber is fully grown, which is to be expected for a time-independent perturbation. We are now interested to see what changes occur when both the bar and SS act together.

As shown by Minchev & Quillen (2006), when two perturbers moving at different pattern speeds are imposed on a galactic disk, an increase in the random motions of stars is expected, i.e., the disk heats. In addition, the authors found that in regions of resonance overlap stars drift radially with time from their birth radii. This suggests that such a mechanism could be responsible for radial migration in the disk. Here, we quantify this idea more clearly for the case of bar + SS by examining the effect of the simultaneous propagation of the two perturbers.

5.1. Time Evolution of ΔL

In Figure 3, we show the time development of the changes of angular momentum in a stellar disk perturbed by both a central bar and SS. In each row, we present a different combination of bar and SS strengths, as described bellow, while all other parameters are kept fixed. As in Figure 2, contour plots show the change in angular momentum, ΔL, as a function of the initial angular momentum L₀ (or radius, r in kpc). Panels from left to right show the temporal evolution of the system for 0.3, 0.6, 1.2, 1.8, and 2.5 Gyr. In the top row, both the bar and the four-armed SS have intermediate strengths: _b = −0.03 (Q_T = 0.25) and _s = −0.02 (Σ_s/Σ₀ = 0.35 for m = 4). For the bottom row, the bar and the two-armed SS are weaker: _b = −0.02 (Q_T = 0.16) and _s = −0.03 (Σ_s/Σ₀ = 0.25 for m = 2). The pattern speeds in both simulations are the same so that we can assess the effect of the different amplitudes.

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Examining the first row of Figure 3, we note that at the beginning of the simulation |ΔL| increases mainly at the CR of each perturber (dotted lines), which would be the case of linearly adding the individual effects of the bar and SS shown in Figure 2. However, at later times large changes in angular momentum occur, suggesting that nonlinear effects become important. Note that at t = 0.6 Gyr the effect near each CR is already stronger than that expected from each individual perturber (compare to Figure 2). Meanwhile, the area between the CRs of the bar and SS fills in with time as a result of the proximity of the 4:1 $\rm ILR_s$, the 4:1 OLR_b, and the 2:1 OLR_b, thus extending the effect over the entire disk. At t = 1.8 Gyr (fourth panel of Figure 3), the changes in angular momentum throughout the disk are similar in amplitude to those resulting from recurrent spirals (Figure 11 in SB02). To reach this level of mixing, however, requires less than ∼2 Gyr compared to ∼9 Gyr for the case of transients.

Decreasing the bar strength to _b = −0.02 (Q_T = 0.16) (bottom row of Figure 3) and considering a slightly weaker two-armed SS has a weaker effect on ΔL. However, changes in ΔL are still very strong, mixing the disk in less than 3 Gyr.

At this time, we can already identify two main differences between radial migration from transients and the mechanism described in this work: (1) the timescale for disk mixing is much shorter in the case of resonance overlap, and (2) resonance overlap causes a bimodal distribution in ΔL with peaks near the bar's CR and its 2:1 OLR, compared to the single peak shown in Figures 11 and 12 in SB02. Next we explore further the effect of perturbation strengths.

5.2. Variation with Perturbation Strengths

The strong increase of |ΔL| with time when both a bar and SS perturb a stellar disk suggests that this is a nonlinear effect. We therefore expect strong variation with the strength of the perturbers. To investigate this further we now explore the effect on mixing when different combinations of _s and _b are considered. In Figure 4, we plot the changes in angular momentum at t = 3 Gyr for simulations with different perturbation strengths, while all other parameters are kept fixed. From left to right the bar strength increases as indicated in each panel. The SS strength increases from top to bottom.

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Observing the top leftmost panel of Figure 4 we see that when both perturbers are weak, mixing is not very strong, although an increase in |ΔL| near the 2:1 $\rm OLR_b$ is still apparent. However, as the bar strength increases, while keeping the spiral weak, the resonance overlap in the inner disk has a surprisingly strong effect on ΔL. To realize this, compare the top row of Figure 4 to the corresponding bar strengths in Figure 2. Similar behavior is observed in the case of a weak bar and a range of SS strengths (first column in Figure 4). Note that here in addition to the inner disk we also see a strong effect near the spirals' CR at 10 kpc, which is well outside the 2:1 $\rm OLR_b$. A possible explanation could be the proximity of the CR of SS to the bar's 1:1 asymmetric, 1:1 symmetric resonances, or x*₁(2) unstable orbits (Dehnen 2000).

5.3. Variation with Pattern Speed Ratio

In Figure 5, we plot ΔL for different SS pattern speeds, Ω_s, while keeping Ω_b the same. This results in different spacing of the bar and SS resonances. From left to right the four-armed SS moves at Ω_s = 0.7, 0.8, 0.9, 1.0, and 1.1 in units of Ω₀ = 30 km s⁻¹ kpc⁻¹. All other simulation parameters are kept fixed: the time is t = 3 Gyr and the strengths of the perturbers are the same as in the top row of Figure 3. As one would expect, the effect of resonance overlap on ΔL is strongest when the highest order resonances (2:1 $\rm OLR_b$ and 4:1 $\rm ILR_s$) lie very close to each other (leftmost panel, L₀ ∼ 7.5). For all pattern speed ratios, the distribution of ΔL exhibits a bimodality near the bar's CR and near its 2:1 OLR. This signature can be used to identify the mechanism described here in fully self-consistent simulations, where the disk dynamics is much more complicated and may include additional mixing due to transient spirals and small mergers.

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We showed in Section 5 that the simultaneous propagation of a central bar and SS in a stellar disk induces radial migration. Since this is a nonlinear effect, depending on the perturbers' amplitudes, strong mixing could occur in a short period of time. Similarly to the majority of disk galaxies, the MW has been well established to be a barred galaxy; thus, we expect mixing due to resonance overlap to be at work here as well. While there is no doubt that the MW has been affected by this process, it may be difficult to estimate exactly how. This migration mechanism is a strong function of the strengths of the MW bar and SS. However, it may be incorrect to use the currently observed spiral and bar amplitudes since most likely they have not been the same throughout the lifetime of the Galaxy. In fact, we expect that spirals were stronger in the past, when the disk velocity dispersion was lower, thus resulting in a more gravitationally unstable disk. In any case, given that the past history of the MW spiral and bar structure is largely unknown, we will make an estimate of how much mixing could have resulted by using the currently observed SS and bar parameters.

6.1. Choice of Bar and Spiral Structure Parameters

6.2. Mixing at Different Galactic Radii

To see in greater detail how stars migrate throughout the disk we now examine the effect on mixing at different galactic radii. To be consistent with MW parameters, we use the simulation for which we presented the changes in angular momentum in Figure 4 (second row, second column). In Figure 6, we show the distribution of birth radii (at t = 0) of stars which are "currently" (at the time indicated in each plot) in an annulus of 400 pc (in the green bin). Each row shows an annulus centered on a different galactic radius, r: from top to bottom r = 2, 4, 6, 8, and 10 kpc. To show the effect on stars placed on nearly circular orbits, the black histograms plot only stars with a cut in the eccentricity, e < 12. We estimate e as $e\approx \sqrt{(u^2+2v^2)/2}$, appropriate for a flat rotation curve (Arifyanto & Fuchs 2006), and u, v are the radial and tangential velocity components of stars with the circular velocity subtracted. The radial velocity dispersion resulting for this sample is ∼5 km s⁻¹, insuring that stellar orbits are almost circular. The red histograms show the full sample. Stars on cold orbits, shown by the (normalized) black histograms, constitute about 10% of the total population.

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First, note that the distributions peak at radii near regions of resonance overlap, as expected. While in the inner disk (r = 2 and 4 kpc) there are strong weights at larger radii, the situation is reversed for r ⩾ 6 kpc. At the solar distance, r = 8 kpc (fourth row in Figure 6), a large population of stars from the bar's CR (r ∼ 4.5 kpc) enters the solar circle. Naturally, when the total population is considered (red histogram) more stars enter from the inner disk due to the exponential profiles of both the density and the radial velocity dispersion. Observe also the overall shift toward larger radii of the cold population compared to the total sample.

While the pattern speed of the MW SS is not well constrained, the bar is well established to have its 2:1 OLR just inside the solar circle. Since the bar's CR and OLR are both strong and roughly invariable (only small changes in the bar parameters are expected in ∼3 Gyr), we can expect to find a bimodal distribution of ΔL with peaks at these bar resonances regardless of the SS pattern speed. That is exactly what we see in Figure 5. What if the MW spirals were transient? In that case, the distribution of the changes in angular momentum can be approximated by the linear combination of a range of simulations with different SS pattern speeds (imagine adding up all panels in Figure 5). This would again result in a bimodal distribution. Therefore, although the past history of the MW structure is not known and a large range of possible perturber strengths, pattern speeds, and longevity exists, it would be reasonable to expect to always find a bimodality in the distribution of ΔL.

Consequently, the large fraction of stars coming from the bar's CR which enters the solar circle (fourth row of Figure 6) may be one of the observational signatures of the spiral–bar resonance overlap mechanism. This is expected to affect the local metallicity distribution of stars. We also note that a large fraction of stars comes from the outer disk, roughly 40% of all stars for this particular pattern speed ratio. This is consistent with the metal-poor stars from the outer Galaxy scattered into the SN, as recently inferred from observational data (Haywood 2008), and a metal-rich population coming from the inner Galaxy. With this new mechanism, the SN stars could have come from a wide range of Galactic disk radii, including from the outer Galaxy, and thus can account for the flatness of the observed AMR.

On the other hand, away from the solar circle we expect to find overdensities of stars originating at the bar's 2:1 OLR. We clearly see this in the initial distribution of stars ending up in the annuli centered on r = 2,  4,  6, and 10 kpc (Figure 6). This may provide another observable for identifying the currently discussed mechanism.

However, the most promising technique to put constraints on this mechanism in the MW is "chemical tagging" (Freeman & Bland-Hawthorn 2002). The idea is to use the detailed chemical abundance patterns of individual stars in our Galaxy to associate them with common ancient star-forming aggregates with similar abundance patterns. This technique will be very important to characterize the star formation and chemical histories of the MW. Chemical tagging will become possible with the HERMES survey on the Anglo-Australian Telescope (AAT; Freeman & Bland-Hawthorn 2008; Freeman et al. 2010) and the APOGEE survey at the Apache Point Observatory (Allende Prieto et al. 2008). Therefore, in the next couple of years we may be in a position to directly measure the spread of open clusters across the Galaxy as a function of age, thus providing direct constraints on the amount of mixing in the MW (Bland-Hawthorn et al. 2010).

6.3. The Effect on Disk Heating

Now we look at the effect on the increase of random motion of stars with time. The observed correlation between the ages and velocity dispersions of SN stars has been a subject of study since the work of Spitzer & Schwarzschild (1951, 1953; see Minchev & Quillen 2006 for a detailed introduction). Deferent mechanisms have been proposed to explain the AVR (or stellar disk heating), including scattering by giant molecular clouds (GMCs; e.g., Mihalas & Binney 1981) and transient spirals (e.g., Barbanis & Woltjer 1967). In Minchev & Quillen (2006), we showed that the resonance overlap of multiple spiral patterns provides yet another mechanism for disk heating. We therefore expect that an overlap of bar and SS resonances would give a similar result. To check if this is indeed the case, in Figure 7 we plot the radial velocity dispersion, σ_r, (solid line) as a function of time for the same simulation used to produce Figure 6. The dashed line shows a power-law fit to our simulated data: σ_r ∼ t^0.29. To compare to observations we also show the AVR in the SN (dotted line), as estimated recently from analysis of the Geneva–Copenhagen Survey data (Holmberg et al. 2009). Note that the observed AVR goes like σ_r ∼ t^0.39. Moreover, in our simulation the disk heats ∼20% more than observed. There could be several possibilities that could account for this discrepancy: (1) our simple model includes neither a gaseous disk component nor star formation, which would naturally lower the velocity dispersion, (2) while here we considered two-dimensional disks, a significant fraction of the heating in the galactic disk plane would be transferred to the vertical direction, where GMCs serve as scattering agents (Jenkins & Binney 1990; Jenkins 1992), and (3) in this work, we used constant parameters for both the MW bar and SS, but it is reasonable to expect more realistic scenarios, where throughout the lifetime of the Galaxy there could have been variations of the perturbers' amplitudes and pattern speeds, bar size, spiral pitch angle, as well as bar destruction and reformation (Bournaud & Combes 2002).

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In this work, we have examined the combined effect of a central bar and SS on the radial migration in galactic disks. We have found that

• 1.  
  This new mechanism for galactic disk mixing could be up to an order of magnitude more effective than the transient spirals one (see Section 5.2). We note that the effect is nonlinear, strongly dependent on the strengths of the perturbers. The nonlinearity of the mechanism is confirmed by the fact that an increase in the amplitudes yields a much larger increase in the angular momentum changes in a simulation including both bar and SS, compared to the case of a single perturber. In other words, the individual effects do not add up linearly. Another indication of nonlinearity is that the effect increases with time only in the case of the two perturbers propagating simultaneously.
• 2.  
  The signature of this mechanism is a bimodality in the changes of angular momentum in the disk with maxima near the bar's corotation and its OLR (see Figures 5 and 6). This is true regardless of the spiral pattern speed and can be used in identifying the spiral–bar resonance overlap migration in N-body simulations, where the disk dynamics is much more complicated.
• 3.  
  For bar and spiral parameters consistent with MW observations we find that it takes ∼3 Gyr to achieve the mixing for which transients require 9 Gyr. In addition to radial mixing, spiral–bar coupling can account for the AVR observed in the SN (Figure 7; Section 6.3). Note however that the estimates for the MW spiral and bar strengths derived from observations now may simply be irrelevant for our understanding of the current state of mixing in the Galactic disk (see Section 6).
• 4.  
  This mechanism could explain observations showing that strongly barred galaxies have weaker metallicity gradients than weakly barred or non-barred galaxies (e.g., Zaritsky et al. 1994; Martin & Roy 1994; Pérez et al. 2009).

The effect of resonance overlap we have described in this paper appears to be quite strong. In order for it to work, a galactic disk must contain a central bar.¹ Unlike in our test-particle simulations, where the disk responds solely to the bar potential, as seen in N-body simulations and deduced from observations, a bar is expected to always drive SS (e.g., Salo et al. 2010). Thus, any barred galaxy would be affected strongly by this mechanism. As the disk heats and SS diminishes in strength, the effect on ΔL will decrease. It is therefore likely that resonance overlap migration will be most vigorous at certain periods during a galaxy lifetime.

In this work, we have only considered steady-state spirals in order to assess the pure effect of the coupling between bar and SS. Considering transient spirals would increase the efficiency, since the SB02 mechanism would come into play as well. Therefore, a combination of a central bar and short-lived transients is expected to be extremely efficient at mixing barred spiral galaxies.

As discussed in Section 6, the most promising technique to put constraints on this mechanism in the MW is "chemical tagging" (Freeman & Bland-Hawthorn 2002; Bland-Hawthorn et al. 2010) which will become possible with the forthcoming spectroscopic survey HERMES (Freeman & Bland-Hawthorn 2008; Freeman et al. 2010). More ongoing and planned large Galactic surveys, such as RAVE, SEGUE, SIM Lite, GYES, LAMOST, and GAIA, can search for signatures of the mechanism (see also Minchev & Quillen 2008).

In a follow-up paper (Minchev et al. 2010b), we study the effect of resonance overlap in fully self-consistent, TreeSPH/N-body simulations including a gaseous component and star formation. The migration mechanism described in this paper can provide an explanation for the observed flatness and spread in the AMR. Our results, however, need to be incorporated into a chemodynamical galactic model to properly assess the effect on the AMR, and possibly also on the formation of the thick disk, similarly to the work by Schönrich & Binney (2009).

We thank Bruce Elmegreen, Ken Freeman, Arnaud Siebert, Jerry Sellwood, and James Binney for helpful communications and discussions. We also thank the anonymous referee for valuable suggestions which have greatly improved the manuscript. Support for this work was provided by Agence Nationale de la Recherche and RAVE.