# Structure in phase space associated with spiral and bar density waves in an N-body hybrid galactic disk

###### Abstract

An N-body hybrid simulation, integrating both massive and tracer particles, of a Galactic disk is used to study the stellar phase space distribution or velocity distributions in different local neighborhoods. Pattern speeds identified in Fourier spectrograms suggest that two-armed and three-armed spiral density waves, a bar and a lopsided motion are coupled in this simulation, with resonances of one pattern lying near resonances of other patterns. We construct radial and tangential () velocity distributions from particles in different local neighborhoods. More than one clump is common in these local velocity distributions regardless of the position in the disk. Features in the velocity distribution observed at one galactic radius are also seen in nearby neighborhoods (at larger and smaller radii) but with shifted mean values. This is expected if the velocity component of a clump sets the mean orbital galactic radius of its stars. We find that gaps in the velocity distribution are associated with the radii of kinks or discontinuities in the spiral arms. These gaps also seem to be associated with Lindblad resonances with spiral density waves and so denote boundaries between different dominant patterns in the disk. We discuss implications for interpretations of the Milky Way disk based on local velocity distributions. Velocity distributions created from regions just outside the bar’s Outer Lindblad resonance and with the bar oriented at from the Sun-Galactic center line more closely resemble that seen in the solar neighborhood (triangular in shape at lower and with a Hercules like stream) when there is a strong nearby spiral arm, consistent with the observed Centaurus Arm tangent, just interior to the solar neighborhood.

## 1 Introduction

The velocity distribution of stars in the solar neighborhood contains structure that has been particularly clearly revealed from Hipparcos observations (Dehnen & Binney, 1998; Famaey et al., 2004). Much of this structure was previously associated with moving groups (Eggen, 1996). Moving groups are associations of stars that are kinematically similar or have similar space motions. Young, early-type stars can be moving together because they carry the kinematic signature of their birth (Eggen, 1996). However, a number of the kinematic clumps identified through the kinematic studies also contain later type and older stars (Dehnen & Binney, 1998; Famaey et al., 2008; Nordstrom et al., 2004; Arifyanto & Fuchs, 2006).

In a small region or neighborhood of a galaxy, structure in the stellar velocity distribution can be caused by different dynamical processes. Bar or spiral patterns can induce structure in the phase space distribution, particularly in the vicinity of Lindblad resonances (Yuan & Kuo, 1997; Dehnen, 1999; Fux, 2001; Quillen & Minchev, 2005; Minchev & Quillen, 2006; Antoja et al., 2009; Gardner et al., 2010; Lepine et al., 2010). Mergers and orbiting satellite subhalos can leave behind stellar streams (e.g., Bekki & Freeman 2003; Meza et al. 2005; Helmi et al. 2006; Gomez et al. 2010). In a local neighborhood, these could be seen as increases in the stellar number density at a particular velocity.

The velocity vector of a star in the solar neighborhood can be described in Galactocentric coordinates with components . Here is the radial component with positive for motion toward the Galactic center. The component is the tangential velocity component subtracted by the velocity of a particle in a circular orbit computed from the azimuthally averaged midplane gravitational potential. The component is positive for a particle with tangential component larger than the circular velocity that is moving in the same direction as disk rotation. The component is the velocity in the direction.

Quillen & Minchev (2005) showed that Lindblad resonances associated with a spiral pattern, given by their approximate velocity component, were likely locations for deficits in the velocity distribution in the solar neighborhood (also see Fux 2001; Lepine et al. 2010). On either side of the resonance the shape of orbits changes. For example closed or periodic orbits can shift orientation so they are parallel to a bar on one side of the resonance and perpendicular on the other side. On resonance there may be no nearly circular orbits. As a consequence in a specific local neighborhood there can be a particular velocity associated with resonance that corresponds only to orbits with high epicyclic variations. These may not be well populated by stars leading to a deficit of stars in the local velocity distribution at this particular velocity. This picture is also consistent with the division between the Hercules stream and the thin disk stars that is likely caused by the outer Lindblad resonance (OLR) with the Galactic bar (Dehnen, 2000; Minchev et al., 2007; Gardner et al., 2010).

Structure seen in the velocity distribution can also be caused by recent large scale perturbations. Perturbations on the Galactic disk caused by bar formation or a galactic encounter, can perturb the action angle distribution. Subsequent phase wrapping can induce structure in the velocity distribution (Minchev et al., 2009; Quillen et al., 2009). Transient spiral density waves can also leave behind structure in the local phase space distribution (De Simone et al., 2004). The local velocity distributions can be particularly complex when both spiral and bar perturbations are present or when multiple spiral patterns are present (Quillen, 2003; Minchev & Quillen, 2006; Chakrabarty & Sideris, 2008; Antoja et al., 2009; Minchev & Famaey, 2010; Minchev et al., 2010b, 2011).

The variety of explanations for structure in the regional phase space distribution presents a challenge for interpretation of current and forthcoming radial velocity and proper motion surveys. We may better understand the imprint of different dynamical structures in the phase space distribution by probing simplified numerical models. Most numerical studies of structure in localized regions of the galactic disk phase space distribution has been done using test particle integrations (e.g., Dehnen 2000; De Simone et al. 2004; Quillen & Minchev 2005; Chakrabarty & Sideris 2008; Antoja et al. 2009; Minchev et al. 2009; Quillen et al. 2009; Gardner et al. 2010). Few studies have probed the local phase space distribution in an N-body simulation of a galactic disk (but see Fux 2000) because of the large number of particles required to resolve structure both in physical and velocity space. In this paper we use an N-body hybrid simulation, integrating both massive and tracer particles, that exhibits both spiral and bar structures to probe the velocity distribution in different regions of a simulated disk galaxy. Our goal is to relate structure in the local phase space distribution to bar and spiral patterns that we can directly measure in the simulation.

## 2 Simulations

In this section we describe our N-body hybrid simulations and their initial conditions. We also describe our procedure for making spectrograms of the Fourier components so that we can identify pattern speeds in the simulations.

### 2.1 The N-body hybrid code

The N-body integrator used is a direct-summation code called -GRAPE (Harfst et al., 2007) that employs a 4-th order Hermite integration scheme with hierarchical commensurate block times steps (Makino & Aarseth, 1992). Instead of using special purpose GRAPE hardware we use the Sapporo subroutine library (Gaburov et al., 2009) that closely matches the GRAPE-6 subroutine library (Makino et al., 2003) but allows the force computations to be done on graphics processing units (GPUs). The integrator has been modified so that massless tracer particles can simultaneously be integrated along with the massive particles (Comparetta & Quillen, 2010). The tracer particles help us resolve structure in phase space without compromising the capability to carry out a self-consistent N-body simulation. The simulations were run on a node with 2 NVIDIA Quadro FX 5800 GPUs. The cards are capable of performing double precision floating point computations, however the computations were optimized on these GPUs to achieve double precision accuracy with single precision computations using a corrector (Gaburov et al., 2009). The Quadro FX cards were designed to target the Computer-Aided Design and Digital Content Creation audience and so were slower but more accurate than other similar video cards even though they did not impliment error correction code.

### 2.2 Initial Conditions for the Integration

The initial conditions for a model Milky Way galaxy were made with numerical phase phase distribution functions using the method discussed by Widrow et al. (2008) and their numerical routines which are described by Widrow et al. (2008); Kuijken & Dubinski (1995); Widrow & Dubinski (2005). This code, called GalacticICS, computes a gravitational potential for bulge, disk and halo components, then computes the distribution function for each component. Particle initial conditions are then computed for each component. The galactic bulge is consistent with a Sercic law for the projected density. The halo density profile is described by 5 parameters, a halo scale length, a cusp exponent, a velocity dispersion, and an inner truncation radius and truncation smoothness, so it is more general than an NFW profile. The disk falls off exponentially with radius and as a sech with vertical height. We have adopted the parameters for the Milky Way model listed in Table 2 by Widrow et al. (2008). A suite of observational constraints are used to constrain the parameters of this model, including the Oort constants, the bulge dispersion, the local velocity ellipsoid, the outer rotation curve, vertical forces above the disk and the disk surface brightness profile.

The number of massive particles we simulated for the halo, bulge and disk are 50,000, 150,000 and 800,000, respectively. The halo is live. The number of test (massless) particles in the disk is 3 million and exceeds the number of massive particles by a factor of 3. The total mass in disk, bulge and halo is , , and respectively. The mass of halo, bulge and disk particles is and , respectively. Snapshots were output every 5 Myrs and our simulation ran for 1.3 Gyr.

The smoothing length is 10 pc and is similar to the mean disk interparticle
spacing. We have checked that the center of mass of the simulation drifts
a distance shorter than the smoothing length during the entire simulation.
The parameters used to
choose timesteps in the Hermite integrator were and
(see Makino & Aarseth 1992; Harfst et al. 2007).
The minimum timestep is proportional to and is
estimated from the ratio of the acceleration to the jerk.
Our value places the minimum possible timestep
near the boundary of recommended practice^{1}^{1}1N-BODY
Simulation Techniques: There’s a right way and a wrong way,
by Katz, N., http://supernova.lbl.gov/evlinder/umass/sumold/com8lx.ps.
The total energy drops by 0.2% during the simulation,
at the boundary of recommended practice that suggests a maximum
energy error^{2}^{2}2http://www.ifa.hawaii.edu/barnes/ast626_09/NBody.pdf
(see Quinlan & Tremaine 1992 on how the shadow distance
depends on noise and softening).

In our goal to well resolve the galactic disk we have purposely undersampled both halo and bulge and so have introduced spurious numerical noise into the simulation from these populations. The product of the mass of halo particles and the halo density density exceeds that of the same product for the disk implying that numerical heating from the massive particles in the disk is lower than that induced by the live halo particles. The large timesteps, relatively small smoothing length and large halo particles imply that this simulation is noisier than many discussed in the literature.

We first discuss the morphology of the simulated galaxy and then use the spectrograms constructed from the disk density as a function of time to identify patterns.

## 3 Morphology

In Figures 1 we show the disk density projected onto the midplane in Cartesian coordinates for a few of the simulation outputs (snapshots). Disk particles are used to create these density histograms, but bulge and halo particles are neglected. In Figures 1 a,b both massive and tracer particles are shown. Figure 1c is equivalent to Figure 1b except only massive disk particles are shown. We find no significant differences between the distribution of massive and tracer disk particles, consistent with the previous study using the same hybrid code (Comparetta & Quillen, 2010).

The numbers of massive particles is sufficiently low that numerical noise is present and amplified to become spiral structure. During the initial 40 snapshots (over a total time of 200 Myr) (shown in Figure 1a) two-armed and three-armed spiral armed waves grow in the region with radius 5–12 kpc. At around a time Myr an open and somewhat lopsided axisymmetric structure grows within a radius of 4 kpc that becomes the bar. At later times both two-armed and three-armed structure exists outside the bar (shown in Figure 1b).

As our halo is live (and not fixed) lopsided motions are not prevented in the simulation. Lopsided motions and three-armed structure has previously been observed in N-body simulations of isolated disks (e.g, see the simulations by Chilingarian et al. 2010). The bulge does not remain fixed during the simulation as a lopsided perturbation develops that moves its center slowly up to a distance of 0.5 kpc away from the initial origin. This distance is larger than the distance our center of mass drifts during the simulation (less than 10 pc). The lopsided motion may be related to the development of three-armed structures in the disk and asymmetries in the bar. Lopsidedness is common in spiral galaxies (see Jog & Combes 2009 for a review), and can spontaneously grow in N-body simulations of disks (Revaz & Pfenniger, 2004; Saha et al., 2007). If the pattern speed is slow, a lopsided mode can be long lasting (Ideta, 2002). There is no single strong frequency or amplitude associated with the lopsided motion in our simulation, though the motion is primarily counter-clockwise and moving with the direction of galactic rotation. We have checked that the center of mass of the simulation drifts less than the smoothing length of 10 pc so the lopsided motion is not due to an unphysical drift in the total momentum in the simulation. However, the growth of the lopsided motion could be due to noise associated with our small smoothing length and large halo mass particles. Simulations of a disk residing in a rigid halo would not exhibit lopsided motions. However as most galaxies display lopsidedness we can consider it a realistic characteristic of a galactic disk. We will discuss this motion later on when we discuss the Fourier component spectrograms.

We also show the disk density in polar coordinates in Figures 2 and 3. These figures show differential density histograms as a function of radius and azimuthal angle () (or in cylindrical coordinates again projected onto the midplane). The bins to create these two plots are logarithmically spaced in radius. The disk density in these two plots has been normalized at each radius. These histogram show where is the mean density at radius averaged over the angle . A logarithmic spiral would give high densities along a straight line in these plots, with slope set by the spiral arm pitch angle. Two armed structures would correspond to two linear but sloped features, each separated by 180. Trailing spiral structures have negative slopes. A bar on these diagrams would correspond to two vertical features separated by 180. The centroid of the galaxy bulge is subtracted prior to computing the density histograms shown in 2 and 3 in polar coordinates but not in Figure 1 that is in Cartesian coordinates. The times of the snapshots in Myr are shown in the upper right hand corners of each panel shown in Figures 1 – 3.

In Figure 2 the differential surface density in polar coordinates is shown at three different times early in the simulation and during the early phase of spiral arm growth. The nearly vertical strip in the leftmost panel tilts and moves to the right at later times. In the right most panel it is a stronger perturbation and with a larger, trailing pitch angle. The spiral arm shown in the center of the leftmost panel becomes more tightly wound, and stronger (as seen on the right) as would be expected from a swing amplification mechanism. After about 50 Myr the spiral patterns are more slowly evolving. Other than during bar formation at about 330 Myr, and in the beginning of the simulation before 50 Myr, the spiral pitch angles do not seem to vary significantly.

In Figure 3 the differential surface density is also shown at three different times but after bar formation. On the lower parts of the panels the bar is seen as pair of vertical strips. We can see that the bar length increases later in the simulation. Exterior to the bar, both two-armed and three-armed patterns are seen and their patterns move more slowly than the bar. The winding or pitch angle of spiral features is about 24 and . This pitch angle exceeds many but not all estimates for spiral arm pitch angles near the solar neighborhood (see Table 1 by Vallée 2008; most pitch angle estimates are ). Other than during bar formation and before 50 Myr, the spiral pitch angles do not seem to vary significantly.

When animated the two-armed and three-armed features appear to constructively and destructively add with density peaks occurring at times and positions when two patterns lie on top of one another (e.g., as discussed by Henry et al. 2003; Meidt et al. 2009). The sum of two different patterns can cause individual arm peaks to shift position and temporarily appear more or less tightly wound.

Kinks or changes in pitch angle of the spiral arms seen in the density images in Cartesian coordinates (in Figure 1b) correspond to places where the spiral arms are discontinuous or change slope in the density distribution as seen in the polar plots (e.g., Figure 3). For example, the spiral pattern just outside the bar has a lower pattern speed than the bar. Consequently the phase of the spiral arms varies with respect to the bar. At times spiral arms appear to be linked with the bar and at other times they appear separated from the bar. Spiral arms in the outer galaxy have slower pattern speeds than those in the inner galaxy. At times the spiral arms appear to be connected and at other times, armlets or kinks are seen.

## 4 Spiral and Bar Patterns in the Simulation

Before we discuss velocity distributions extracted from local neighborhoods we first discuss patterns of waves measured from the simulations.

### 4.1 Constructing Spectrograms

We constructed spectrograms using the procedure outlined by Meidt et al. (2008) in their section 3.2 (also see section 3.4 by Masset & Tagger 1997). In each radial bin, defined by a binning function , and at each snapshot time we compute

(1) |

where the sum is over all disk particles in the radial bin (both massive and tracer) and the integer, , is the azimuthal wave number. Bulge and halo particles are ignored. We use logarithmically spaced radial bins. We compute the Fourier transform of the complex function composed of and or

(2) |

where is a Hanning function spanning our time window between snapshot time and . In our spectrogram figures we show the amplitude of the complex Fourier components or . In our spectrograms the angular frequency, , increases along the -axis and increases along the -axis).

A bar or two-armed spiral structure has strong Fourier components in its density distribution (calculated with equation 1). The pattern speed of the wave is related to its angular frequency by . Pattern speeds are found by dividing the angular frequencies of horizontal features in the spectrograms by the integer .

During an N-body simulation with a non-stationary or live halo, lopsided modes or waves may develop and the bulge and central disk of the galaxy may not remain at a fixed position. If the Fourier components are measured respect to a radial position that is not centered on the bulge centroid then an component can generate other Fourier components. This is sometimes called ‘aliasing’ or ‘mode-mixing’ (e.g., Olling & Dehnen 2003). As a consequence we subtract the position of the centroid of the galaxy bulge prior to computing the Fourier coefficients with . We will also discuss the spectrogram computed without subtracting the bulge centroid position.

### 4.2 Pattern speeds

After Myr the bar becomes stable and then evolves more slowly during the rest of the simulation. Well defined frequencies are seen in spectrograms after bar formation. Figures 4 shows spectrograms constructed from Fourier components, whereas Figure 5, 6 and 7 show spectrograms constructed from , 3 and 1 Fourier components, respectively. The spectrogram from Figure 4a is measured from the middle of the simulation whereas 4b is measured from the second half of the simulation. The same time periods are used for Figures 5 – 7. We can see from a comparison of Figure 4a and Figure 4b (also see Figures 1 and 3) that the bar grows longer and its rotation rate slowly decreases during the simulation. At 400 Myr the bar is only 6 kpc long (semi-major axis) but by 700 Myr its semi-major axis has increased to 11 kpc.

Pattern speeds of waves can be estimated from a spectrogram by dividing the angular frequency of a strong feature by the integer used to make the spectrogram. In our spectrograms we plot the angular frequency, , rather than the pattern speed , following most previous works (e.g, Sellwood & Sparke 1988; Masset & Tagger 1997). Estimated pattern speeds for waves seen in the presented spectrograms are listed in Table 1. The strongest pattern seen in the spectrogram shown in Figure 4a is the bar wave that has an angular frequency of Myr corresponding to a pattern speed of Myr. There are slower patterns at larger radii seen both in the middle and later parts of the simulation. These correspond to two-armed spiral patterns. Figure 6 showing the spectrograms shows that there is also at least one strong three-armed pattern present in the simulation.

Figure 4 and 5 show curves corresponding to the angular frequencies with and , respectively where is that used to compute the spectrogram. Each pair of lines (when there is a in the above listed frequencies) is the same color in the Figures. These lines can be used to identify inner and outer Lindblad Resonances. Figure 6 and 7 in addition show curves corresponding to the angular frequencies , with and , respectively. Here and are the angular rotation rate and the epicyclic frequency. We computed these frequencies using a rotation curve generated from the azimuthally averaged midplane gravitational potential measured from the positions of all the massive particles at time Myr.

A Lindblad resonance (LR) with a pattern with speed, , and integer occurs where

(3) |

(Lindblad, 1926). The sign in the above determines whether the resonance is an inner or outer Lindblad resonance (ILR or OLR). A corotation resonance (CR) occurs where , or the angular rotation rate equals the pattern speed. Here we refer to each resonance by its integer, here denoted . Other works refer to the ILR as the 4:1 ILR (e.g., Meidt et al. 2009) or the 4/1 ILR (e.g., Rautiainen & Salo 1999). We also use the notation ILR to refer to the ILR. The radii of the estimated Lindblad and corotation resonances for pattern speeds estimated from the spectrograms are also shown in Table 1. Resonance locations are estimated using the rotation curve computed from the azimuthally averaged density distribution at time Myr. The bar pattern also contains non-zero Fourier components in its density distribution with . As the bar lacks symmetry when rotated , or when reflected about its major or minor axis, it also contains non-zero odd- Fourier components in its density distribution.

We see in Figure 4a that the bar pattern extends out to its corotation radius where the bar pattern intersects (the dark blue line) at about 6 kpc. The pattern is seen past the bar’s corotation radius in the spectrogram, consistent with previous studies (e.g., see Figure 3 by Rautiainen & Salo 1999). Between the corotation resonance and the bars outer Lindblad resonance (OLR), periodic orbits can be elongated and oriented perpendicular to the bar.

Recent studies of the Milky Way have revealed two bar like structures, a long thin bar (Benjamin et al., 2005) at about from the Sun/Galactic center line and half length 4.4 kpc and a shorter or traditional thicker bar (or triaxial bulge component) at about (Vanhollebeke et al., 2009). Estimates of the bar’s pattern speed are in the range 50-55 km s kpc or Myr (Dehnen, 2000; Minchev et al., 2007; Gardner et al., 2010). As shown by Gardner et al. (2010) the Hercules stream is consistent with either long or short bar as long as the pattern speed places the solar neighborhood just outside the bar’s OLR. Our simulated bar has with half length kpc and so is longer than the Milky Way’s long bar but this is consistent with its slower pattern speed at 0.040 Myr or 40 km s kpc. In our simulation a position corresponding to that of the solar neighborhood would be at a galactocentric radius of about 10 kpc, just outside the bar’s OLR.

At both earlier and later times, features at lower pattern speeds are also seen in the spectrograms. A peak just past the end of the bar is seen with a pattern speed of Myr at earlier times (Figure 4a) and at at later times (Figure 4b). This would correspond to an inner two-armed spiral pattern. This pattern appears to end within its own corotation radius at 7 or 9 kpc (early and late, respectively). There is also a slower outer two-armed spiral pattern primarily seen at early times with a pattern speed (though possibly also at later times with ). This pattern also ends near its own corotation resonance.

There is a weak and narrow higher frequency peak in the spectrograms near the end of the bar at an angular rotation frequency of and Myr at early and late times, respectively. Unlike the slower frequency in the spectrograms that end near their corotation radii, this one extends between its corotation radius and its outer Lindblad resonance. There is a similar higher frequency feature in the spectrograms (Fig. 5) at and 0.17 Myr at early and later times, respectively.

The spectrograms shown in Figure 5 look remarkably similar to the spectrograms (Figure 4). Two armed and oval density distributions often contain components so this is not necessarily surprising. The higher frequency features near the end of the bar are also seen in the spectrograms. With the exception of the bar itself, the peaks in the spectrogram extend between their inner and outer Lindblad resonances (shown as green lines). This differs from the two-armed spiral patterns that end within their corotation radii. In the spectrogram the inner spiral pattern appears split into two different peaks. We may be able to better measure angular frequencies in the spectrogram than the spectrogram because the angular frequencies are higher and so less affected by the precision limit set from the width of the time window used to compute the spectrograms.

Unlike the inner and outer two-armed patterns, the three-armed pattern both at early and later times extends through its corotation resonance. This pattern (see yellow lines in the spectrograms shown in Figure 6) extend between its inner and outer Lindblad resonances. The pattern speed is difficult to measure at early times because there are two nearby features in the spectrogram. Both at early and later times the pattern speed of the three-armed spiral is about Myr.

As noted by Shaviv (2003) (see their Table 3) pattern speeds estimated for spiral patterns in the solar neighborhood fall into two groups. One group has pattern speed km s kpc similar to the angular rotation rate of a particle in a circular orbit, , at the solar galactocentric radius, , and the other group has a pattern speed km s kpc. The mock solar neighborhood position in our simulation lies outside the inner two armed spiral pattern at 0.030 Myr, but is affected by the three armed spiral wave with pattern speed 0.023 Myr or km s kpc and the outer two armed spiral with a pattern speed of 0.015 Myr or km s kpc. If the spiral waves present in the solar neighborhood are driven by the Galactic bar then it might be interesting to match the faster pattern speed measured in the solar neighborhood with a three armed pattern driven by the bar and the slower pattern speed with a two armed pattern also possibly linked to the bar. However, with the exception of Naoz & Shaviv (2007), few studies have considered interpretation of the solar neighborhood in terms of two or more patterns.

Previous works have seen spiral structure in N-body or particle mesh simulations at pattern speeds that differ from the bar’s (e.g., Sellwood & Sparke 1988; Patsis & Kaufmann 1999; Rautiainen & Salo 1999; Voglis et al. 2006). Masset & Tagger (1997) interpreted this behavior in terms of non-linear mode coupling (Tagger et al., 1987; Sygnet et al., 1988), whereas Sellwood & Lin (1989) interpret the differences in pattern speeds in terms of transient behavior and a groove instability. The spiral patterns in the simulations by Sellwood & Sparke (1988) extend from their inner Lindblad resonances to their outer Lindblad resonances (see their Figure 1). However, the patterns of the two spiral density waves seen by Patsis & Kaufmann (1999) in a simulation lacking a bar extend from their inner to their outer Lindblad resonances. Both of these situations differ from the two-armed structures in our simulations that end within their corotation radii. The simulations by Rautiainen & Salo (1999) are closest in behavior to ours, showing multiple spiral patterns extending past the bar, with some patterns ending near or within their corotation radii.

As the spiral and bar patterns in our simulations are moderately long lived, the spiral patterns could be driven by the bar (Yuan & Kuo, 1997; Masset & Tagger, 1997; Rautiainen & Salo, 1999). Rautiainen & Salo (1999) accounted for the ranges and speeds of spiral patterns using the non-linear wave coupling scenario. They found that they could often associate the beginning of a slower outer pattern with the location of one of its resonances that is also a resonance with an inner faster bar or other spiral pattern. The overlap between the resonances relates one pattern speed to the other and couples one perturbation to the other. Rautiainen & Salo (1999) found examples of CR/ILR, CR/ILR, OLR/CR and OLR/ILR couplings between spiral and bar waves and between two different spiral density waves.

For our system there is a possible connection between the bar’s CR and the ILR of the three-armed spiral armed wave. A comparison of resonance locations (see Table 1) shows that the bar’s corotation resonance lies close to the three-armed spiral’s ILR. The radial distance between these resonances is small compared to the error from measurement in the spectrogram. There is also a possible coupling between other resonances. It is difficult to differentiate between possibilities by resonance locations alone as driving could occur in a spatially broad region near a resonance. As suggested by Masset & Tagger (1997) we can test for mode or wave coupling by searching for waves at beat frequencies though Rautiainen & Salo (1999) found that expected beat waves are not always detectable.

### 4.3 Non-linear Coupling of Waves

The scenario introduced by Tagger et al. (1987) suggests that there are strong non-linearities in the stellar response when resonances coincide. Two waves (e.g., bar and spiral) couple to a third one at a beat frequency. For example we consider azimuthal wavenumber associated with a bar perturbation at pattern speed and so frequency . We consider azimuthal wavenumber associated with a spiral pattern at pattern speed and so angular frequency . A third wave can be excited with an azimuthal wavenumber and angular frequency or with wavenumber and .

An example explored by Tagger et al. (1987) is a bar and spiral mode, both with , coupled to a slow axisymmetric mode with and a faster four-armed one with (Masset & Tagger, 1997). Like Masset & Tagger (1997) we see features in the spectrograms at frequencies above that of the bar, and these can be considered predictions of the wave-coupling scenario. Unfortunately the number of features present in the and spectrograms is high and the imprecision of measured frequencies makes it difficult to definitively determine whether there is coupling between waves and a slow axisymmetric mode.

As our simulation contains a bar (), a lopsided motion and a three armed wave we can consider coupling between these three waves. To create the , 3 and 4 spectrogram we first subtracted the bulge centroid, however the galaxy center moves during the simulation. In Figure 7, showing the spectrograms but constructed in the inertial frame, we see features associated with the major patterns identified in the other spectrograms. For example we see a peak at Myr that coincides with the bar’s angular frequency. One interesting peak is at an angular rotation rate of that is also seen in the spectrogram at a radius of about 8 kpc (). At earlier times there is power at in the spectrogram that is broader than the similar feature seen in the spectrogram. This suggests that there is sufficient power at low frequencies in the lopsided wave that it could couple to other waves.

Since there is power at low frequencies in lopsided motion, we can consider the possibility of a coupling between a slow lopsided mode and the bar. Here the index and pattern speed of the lopsided wave is and that of the bar is . The sum leads to a three-armed spiral pattern with angular frequency or pattern speed . If we assume that this corresponds to the three-armed pattern with then we can solve for the angular frequency of the lopsided mode . There may be power at this frequency as there are low frequency features in the spectrogram. Thus, a lopsided slow mode could be coupling the bar and the three armed spiral waves.

A variety of resonant overlaps or non-linear wave couplings are possible and may account for the driving of the patterns present in the simulation. If we could more precisely measure frequencies of the patterns then we would better determine which associations are most important or likely. Unfortunately peaks in the spectrograms are sensitive to the range (in time) of simulation data used to construct the spectrogram (e.g., compare Figure 4a to Figure 4b). Longer time series are required to make more accurate frequency measurements. Since the pattern speeds are not fixed, increasing the size of the time window does not increase the precision of a frequency measurement. This is particularly the case for slow patterns that might be important for the non-linear coupling models.

In summary, we find that the spectrograms are rich in small features. The shape of the spectrograms seen at early times resembles that at later times implying that the frequencies of these smaller features are slowing along with the bar pattern. Even though there are many waves present, they are not stochastically appearing and disappearing but slowly evolving. Lindblad resonances associated with one wave are likely to be coincident with resonances of another wave present in the simulation. Together these findings suggest that the waves present in the simulation are coupled due to non-linear interactions. The most convincing and strongest coupling is that between the bar, the lopsided wave and the three-armed spiral wave. The multitude of small features present in the spectrograms suggest that additional couplings are present. The presence of features in the spectrograms at frequencies above that of the bar is consistent with the wave-coupling scenario.

## 5 Structure in the Plane

We first describe how we construct velocity distributions in local neighborhoods. We then discuss possible relations between features seen in these velocity distribution and the spiral or bar patterns.

### 5.1 Velocity distribution histograms in local neighborhoods

We construct velocity distribution histograms in different local neighborhoods in the following manner. Consider a neighborhood centered about a point in the disk plane with coordinates or in cylindrical coordinates . We will refer to neighborhoods by the cylindrical coordinates of their centers; (). We consider a star or simulated particle in the neighborhood of this center point if its coordinates have distance in the plane, where is a dimensionless scaling factor setting the size of the neighborhood. We compute the radial and tangential components of the velocity of each star with respect to the vector to the galactic center defined by the central coordinates of the neighborhood, . The component is times the radial velocity component in this galactic coordinate system. The component is the tangential component in this coordinate system minus the circular velocity estimated from the azimuthally averaged midplane rotation curve computed at time Myr.

Histograms showing the velocity distribution are computed from the disk particles in each neighborhood. The computed histograms are effectively coarse-grained phase space distributions. The larger the neighborhood, the larger number of stars in the neighborhood. If there are too few stars in the neighborhood then the velocity distribution is noisy. If the neighborhood is large then structure present on small spatial scales may be masked or smoothed by the coarse graining. An extreme example would be if the neighborhood were so large that it contained two spiral arm density peaks.

A series of velocity distributions ( histograms) is shown in Figure 8 at simulation time 600 Myr. Velocity distributions are extracted from neighborhoods centered at 8 equidistant angles and at 6 radii spaced logarithmically. Each row contains histograms from neighborhoods at a particular galactocentric radius. The lowest row has and the top row . Each radius differs from the previous by a factor of 1.25. Each column shows neighborhoods with a particular galactocentric angle with the leftmost column aligned with the bar. Each neighborhood is separated in angle by 45 with angle increasing counter-clockwise in the galaxy plane. Each neighborhood has a size set by the factor setting the maximum distance of stars included in the neighborhood from its center. For the neighborhoods shown in Figure 8 the ratio of the neighborhood radius to is . Positions and sizes of each neighborhood are shown in Figure 9 in both Cartesian and polar coordinates. At kpc the radius of the neighborhood corresponds to 0.8 kpc which is small compared to some N-body studies (e.g., Gomez et al. 2010 who used a neighborhood radius of 2.5 kpc) but larger than the distance of stars in solar neighborhood samples (e.g., Siebert et al. 2011).

We have compared the velocity distributions created with and with with smaller neighborhoods. A comparison at time 800 Myr and at a radius of 10 kpc is shown in Figure 10. We also show a velocity distribution constructed of massive disk particles only to illustrate that we see no evidence for significant differences between massive and massless disk particle populations. The velocity distributions in smaller neighborhoods are noisier as would be expected since each neighborhood contains fewer particles. The clumps in the plots are slightly sharper, otherwise the morphology is almost identical. The density distribution is relatively smooth (see Figure 9) implying that the velocity dispersion is not low in the disk (though it is lower at larger radii than at smaller radii). A simulation of a colder disk with more tightly wound spiral arms might show features in the velocity distributions that are more strongly dependent on the size of the neighborhood.

In Figure 8 we see that multiple clumps exist in the local velocity distribution at all radii in the galaxy. Velocity clumps were previously seen in a local neighborhood of the N-body simulation presented by Fux (2000). Since our simulation lacks mergers, the clumps in the velocity distribution are caused purely by dynamical phenomena in the disk (as also concluded by Fux 2000). Clumps in the velocity distribution in one neighborhood are often also present in the nearby neighborhood at a larger or smaller radius but at a shifted velocity. For example at an angle of 315 there is an arc with a mean velocity of km/s for neighborhood radius that has a lower velocity of km/s for at . We can consider the possibility that a single halo particle could be responsible for one of these features. The separation between the two neighborhoods mentioned above is a kpc. However the circular velocity at a distance of a kpc from one of our halo particles is only 2 km/s. Thus a single halo particle passing through the disk cannot account for the correlated and broad structures seen in the velocity distributions. We will discuss the relationship between and the mean orbital or guiding radius below when we consider the relation between velocity components and orbital properties.

### 5.2 Velocity distributions in arm and interarm regions

When looking at the velocity distribution in different neighborhoods it is helpful to know which neighborhoods have a high surface density. Figure 9a and b also show the locations of neighborhoods on the projected disk surface density used to construct the velocity histograms shown in Figure 8. In Figure 9b, projected in polar coordinates, the axis gives the azimuthal angle with respect to the bar. Thus each neighborhood shown on this plot directly corresponds to the position of the histogram in Figure 8. This makes it possible to see how the velocity distribution varies with position and between arm and interarm regions.

A comparison between Figure 8 and 9b allows us to see if the velocity distribution in high density regions differs from that at low density regions. We find that when the density increases the velocity dispersion is also likely to increase. The velocity distribution is wider when the neighborhood lies on an arm or bar peak than in interarm regions. Phase space density should be conserved as particles orbit the galaxy, so this trend is expected.

We can compare the velocity distribution on the leading (or pre-shock) side of a spiral arm to that on the trailing (or post-shock) side. Our simulated galaxy rotates counter-clockwise or with increasing . We denote the leading side of a spiral arm as that located at larger than the peak, or to the right in Figure 9b (and counter clockwise in angle from an arm peak in Figure 9a). Observed galaxies tend to have dust lanes on the trailing side (as denoted here) of a spiral arm. This happens because the spiral pattern is usually faster than local galactic rotation (in other words the corotation radius lies interior to the pattern). In a frame moving with the gas on Figure 9b, the pattern moves to the right and so shocks would form on the left-hand or trailing side of each density peak.

We see that arcs in the velocity distribution with positive and are often seen on spiral arm density peaks and on the leading (or pre-shock) side of an arm. These arcs are oriented with negative slope such that increases with decreasing . Moving from the leading to the trailing side at a particular radius (comparing histograms from right to left in Figure 8) the high arcs move from positive to negative . See for example and or and in Figure 8. Interarm regions tend to lack arcs that are centered at high and positive .

In interarm regions the velocity dispersion is lower and there is sometimes a clump, separated from the dominant peak, that is at negative and . The clump can be elongated in , resembling the Hercules stream in the solar neighborhood’s velocity distribution (e.g., Dehnen & Binney 1998; Arifyanto & Fuchs 2006; Gardner et al. 2010; Bovy 2010).

### 5.3 Gaps in the histograms

Figure 12 shows the neighborhoods at a radius of 6.4 kpc for the same angles shown in Figure 8 but for snapshots separated by 50 Myr between 500 and 950 Myrs. The angles here are given with respect to the bar and so in a coordinate system rotating with the bar. At each angle with respect to the bar, clumps do not remain fixed as the system evolves. This implies that spiral structure influences the locations of gaps and clumps in the velocity distribution. The gaps cannot be solely due to bar perturbations.

A stream similar to the Hercules stream at km/s is present in the velocity distribution at early times at angle 0 (aligned with the bar) at kpc. Like the Hercules stream (e.g., Dehnen 1999) this arc or clump has a wide distribution, a narrow distribution and is centered at a negative value. However the bar’s OLR (at about 9 kpc) lies outside these neighborhoods, so this clump cannot be associated with this resonance, Similar feature low clumps in the velocity distribution are also seen at other radii such as and 10 kpc but appearing at different times in the simulation and at different angles with respect to the bar, consequently the bar and resonances associated with it cannot provide the only explanation for these features.

The spiral density waves have slower patterns than the bar, hence density peaks associated with them move to the left (or to a smaller ) as a function of time or moving downward on Figure 12. At one position as a wave peak passes through the neighborhood first a velocity distribution typical of a leading side of an arm is seen then one typical of a trailing side. The velocity distribution contains a positive and feature when on the leading side of a spiral feature. The feature is centered at progressively lower values as the arm passes by and the neighborhood moves to the trailing side. See for example the right most column and at an angle of 315 between times 500 to 700 Myrs (top 5 rows).

While clumps come and go in time, we find that the gaps between the clumps tend to remain at the same values. For example consider the right most column of Figure 12. Two gaps are seen with different degrees of prominence at differ times. We recall from Figure 8 that gaps at one neighborhood radius were often seen at another neighborhood radius but shifted in . To compare a gap seen in a neighborhood at one radius to that seen in a neighborhood at another radius we can consider motions of particle orbits.

## 6 Interpreting structure in the plane

To relate structure in local velocity distributions it is helpful to recall models for orbits. We first consider epicyclic motion in the absence of perturbations, then review the orbital dynamics to first order in the perturbation strength in the presence of perturbations. In both case we consider the relation between velocities in a local neighborhood and quantities describing the orbit such as a mean or guiding radius and an epicyclic amplitude.

In the absence of perturbations from spiral arms, the motion of stars in the disk of a galaxy can be described in terms of radial or epicyclic oscillations about a circular orbit (Lindblad, 1926; Kalnajs, 1979). It is useful to specify the relation between the observed velocity components and the parameters describing the epicyclic motion (e.g, Fux 2000, 2001; Quillen & Minchev 2005) or the mean radius or guiding radius, , and the epicyclic amplitude. The energy of an orbit in the plane of an axisymmetric system (neglecting perturbations from spiral structure) with a flat rotation curve is

(4) |

where the potential energy, , is that appropriate for a flat rotation curve, is the Galactocentric radius and is the circular velocity.

With an epicyclic approximation we can write the energy

(5) |

where the term on the left is the energy of a star in a circular orbit about a guiding radius and is the energy from the epicyclic motion,

(6) |

Here is the epicyclic amplitude and is the epicyclic frequency at the guiding radius .

We now consider stars specifically in a neighborhood of galactocentric radius , restricting us to a specific location in the galaxy. Setting the energy equal to that written in terms of the epicyclic motion using equations (5, 6), we solve for the guiding radius, . It is convenient to define the distance between the guiding or mean radius and the neighborhood’s galactocentric radius, . To first order in and we find that a star in the neighborhood with velocity components has a guiding radius with

(7) |

and epicyclic amplitude

(8) |

The factor of a half is from equivalent to 2 when the rotation curve is flat. An angle describing the phase in the epicycle is

(9) |

(consistent with equations 2-4, Quillen 2003, for a flat rotation curve). The epicyclic phase angle at apocenter.

The above three relations allow us to relate the velocity components for stars in the solar neighborhood, to quantities used to describe the epicyclic motion; the guiding radius, epicyclic amplitude and phase angle. Particles with positive have and so guiding radii that are larger than the radius of the neighborhood, . Particles with negative have and so are expected to spend most of their orbits inside . The distance from the origin, , determines the epicyclic amplitude.

It may also be useful to write

(10) |

where we have used our definition for in equation 7. This equation shows that to first order sets the guiding radius of a particle. This is useful because the guiding radius sets the approximate angular rotation rate and rotation period of the orbit. The guiding radius also sets the approximate epicyclic frequency (or period) of the particle’s orbit. Resonances occur where sums of integer multiples of these periods are equal to an integer multiple of a pattern speed. Thus the value of clumps or gaps in the velocity distribution may also allow us to locate resonances.

### 6.1 Spiral Perturbations to First Order in the plane

The above approximation neglects the effect of bar or spiral perturbations. Spiral waves give a time dependent perturbation to the gravitational potential that can be approximated by a single Fourier component depending on a single frequency. For a logarithmic spiral perturbation with arms we can consider where the pitch angle . To first order in the perturbation strength the orbit of a star in the galactic midplane with mean or guiding radius

(11) | |||||

(Binney & Tremaine 1987; equation 3-119a) where is an epicyclic amplitude, and are phase angles. The forced epicyclic amplitude in the WKB approximation where represents the distance to a Lindblad resonance and is evaluated at the guiding or mean radius, and and are the epicyclic frequency and angular rotation rate at . The denominator becomes small near the Lindblad resonance and the first order approximation breaks down. A higher order approximation can be used to show that closed orbits exist on both sides of resonance and that epicyclic amplitudes do not become infinite on resonance (e.g., Contopoulos 1975).

The orbital motion (equation 11) consists of an epicyclic motion with amplitude and a phase and a forced motion aligned with the potential perturbation and moving with it. A population of stars would have a distribution in both and . When and the orbits are ellipses. The orbit with is periodic or closed in the frame rotating with the spiral pattern. The mean value, , sets the velocity dispersion in the disk. When the orbits are nearly closed in the frame rotating with the potential perturbation and we can consider the disk to be cold.

The radial and tangential velocity components can be computed

(from equation 3.117 by Binney & Tremaine 1987). When the rotation curve is flat is independent of radius and the velocity components and .

We note that the equations for and contain similar terms but with the opposite sign. A shift in corresponds to a shift in radius and the mean orbital radius can be roughly estimated from the component as was true in the case of pure epicyclic motion (equation 10). If the free epicycle is small () or the disk is cold then

(14) |

with

(15) |

and is sensitive to the angle with respect to the potential perturbation (compare to equation 9) for pure epicyclic perturbations). Likewise the distance from origin on the plane is dependent on the amplitude of radial oscillations with

(16) |

(compare to equation 8).

In the limit , particles are in nearly elliptical orbits but the orientation of the ellipses can vary with radius. A set of ellipses with differing orientations is shown in Figure 11a. Each ellipse has the same eccentricity but is shifted in angle with respect to the previous one. The difference in angle is 20 except between the 6th and 7th ellipse. The difference in orientation angles between these two ellipses is larger, 90, and introduces a discontinuity that we will discuss later when we consider the relationship between gaps in the velocity distribution and discontinuities in the spiral structure. The red circle on Figure 11a shows an example of an arm peak that lies outside the discontinuity. In this neighborhood there are nearly overlapping orbits. The range of velocities seen in this neighborhood would be high as there are stars from many different orbits passing through the neighborhood. There should be a relation between and velocity components as there is a relation between mean galactic radius of the ellipse and orbital angle in the neighborhood. Thus the velocity distribution would exhibit an arc.

The green circle in Figure 11a shows an interarm region where the angles of orbits only varies slightly compared to that in the red neighborhood. In the green interarm neighborhood we expect a narrow velocity distribution. This illustrates why we see expect to see arcs in the velocity distributions on spiral arm peaks and lower velocity dispersions in interarm regions.

An arc in a local velocity distribution that has decreasing as increases can be interpreted in terms of orbital properties as a function of (approximately set by ). The positive orbits have larger guiding radii. The change in angle or slope on the velocity distribution plots implies that the epicyclic angle (equation 15) decreases with increasing . Thus these arcs correspond to a relation between epicyclic angle and guiding radius. Consider our cartoon, Figure 11a, showing concentric ellipses with different orientation angles. A neighborhood on top of a spiral arm peak (such as the red one) intersects orbits with a range of epicyclic angles, corresponding on this figure to a smooth change in orientation of the ellipse. Thus we expect that an arc would be observed in a velocity distribution at the position of a spiral arm peak. The angle of the arc in the velocity distribution is likely to be affected by the winding angle of the spiral structure with a more open spiral corresponding to a arc with steeper slope. However a neighborhood may contain orbits with a smaller range of guiding radii if the spiral structure is less tightly wound. Model orbital distributions would be needed to use the slopes and ranges of velocities of clumps seen in velocity distributions, and their gradients, to place constraints on the spiral structure (pattern speed, pitch angle and strength) from observations of the velocity distribution alone.

We now consider the type of arc that would be seen on a trailing spiral arm. The red neighborhood in Figure 11a intersects orbits near apocenter with small guiding radii (and so negative ) and orbits that are near pericenter that have larger guiding radii (and so positive ). In between and in the neighborhood are orbits that are moving toward the Galactic center and so have positive values. For trailing spiral structure we expect the arc to progress from low to high passing through the top right quadrant of the plane, as we seen in arm peaks in Figures 8, 12 and 13. Thus our illustration Figure 11a provides an explanation consistent with spiral structure for the arcs seen in the velocity distribution.

We can use our first order perturbation model (equations 11 - 6.1) to consider particles with nearby guiding radii. We expand equation (11) for closed orbits ()

where we have used the WKB approximation by neglecting . The derivative of the above equation

(18) |

Consider two closed orbits separated by a small difference in guiding radius. The radial distance between these closed orbits at a particle angular location is smallest when the above derivative is a minimum. A small difference in current radius is possible when the sine term is negative. This happens midway between pericenter and apocenter. Thus the closed orbits are nearest to each other when this sine term is negative (). The radial velocity is also proportional to the sine of the angle. For , corresponding to trailing arms, we find that the orbits are closest together when the radial velocity component is negative and the particles are approaching pericenter (positive , zero ), as shown by the red circle in Figure 11a. Orbits are maximally distant when the radial velocity is positive or 90 degrees away from the red circle on Figure 11a that shows a two-armed structure. At locations and corresponding velocities where the above derivative is high, the phase space density would be higher and so there would be features seen in a local velocity distribution. Inclusion of the the derivative when estimating the derivative would shift the estimated angle of the minimum of the derivative. This shift may explain why arcs in the velocity distribution are primarily located in the upper right hand quadrant (positive ) and not centered at , .

The above discussion has neglected the dispersion of orbits about the closed or periodic ones shown in our cartoons (Figure 11), however if the dispersion is neglected then the predicted velocity distribution contains particles at only a few velocities. The stellar dispersion has been used in constructing weights for orbits to populate the velocity histograms and predict velocity distributions from test particle integrations (e.g., Dehnen 2000; Quillen & Minchev 2005). We should remember that Figure 11 only qualitatively gives us a relation between orbital and velocity distribution, however the properties of closed orbits are relevant for interpretation of these distributions (e.g, Quillen & Minchev 2005).

The above equations are given in the WKB approximation for a spiral perturbation. For a bar or two-armed open spiral perturbation to first order the orbits are also ellipses however the WKB approximation cannot be used. For a bar perturbation an additional term in should be taken into account when estimating the guiding radius and the angle from (see Binney & Tremaine 1987 section 3.3). For a spiral perturbation if the WKB approximation is not used additional terms in both velocity components would be used to estimate the guiding radius, the and orbit orientation and the angle with respect to it from (e.g., Contopoulos 1975).

## 7 Gaps at different radii

We now look at the location of gaps in the velocity distribution for the snapshot shown in Figures 8 and 9 at Myr. In Figure 8 at a radius of kpc and an angle of there are two gaps seen in the velocity distribution, one centered at and the other at km/s. The one at has a guiding radius of kpc, the same as the neighborhood radius. Using equation 10, we can predict the value for this gap were it to be observed from at a neighborhood radius of kpc. We estimate that it should lie at 50 km/s. There is an arc with a gap at about km/s in the velocity distribution at kpc and at , as expected. The other gap in the neighborhood with at km/s has a guiding radius kpc (again using equation 10). A clump with this guiding radius would lie at km/s in a neighborhood with kpc. We don’t see a gap with this value at kpc and but we do see one in the neighboring position at kpc, . Thus we find that relation between and guiding radius is consistent with shifts in individual features in as a function of neighborhood radius.

We also find gaps at nearly zero values in neighborhoods with and 10 kpc. In the inner galaxy there is a gap at kpc, with slightly negative that is also present at km/s in the neighborhood with kpc, and has kpc. Altogether we see gaps at the following guiding radii 3.9, 5.1, 6.3, 8 and 10 kpc at this time in the simulation.

Looking back at Figure 8 we compare the location of gaps in the velocity distribution (identified by their guiding radii) and the locations of kinks (changes in winding angle) or discontinuities in the spiral arms. In Figure 9b we have identified radii at which we find discontinuities in the spiral structure, either changes in winding angle (kinks) or places where spiral arms or armlets appear or disappear. These radii are shown as black circles in Figure 9a and as horizontal lines in figure 9b. The radii of the discontinuities on these plots are at , 5.1, 6.3, 8.7 and 11 kpc. A comparison between the list of discontinuities and our list of gaps identified in the velocity distributions (identified by their guiding radii) suggests that these two lists are related.

A gap in the velocity distribution occurs in a neighborhood when there are two groups of stars each with velocity vectors in similar but different directions, but few stars with velocity vectors in between those of the two groups. In Figure 11a we show concentric ellipses that increase in radius and shift in orientation angle. The difference in orientation angle between each ellipse is small and constant except between the 6-th and 7-th ellipse where the difference is much large and 90. There is a discontinuity in the position of the spiral arm peak at the radius of the introduced large change in orientation angle that we can see from the shape of the orbital overlap regions. The discontinuity introduces regions such as the neighborhood illustrated with the blue circle. This neighborhood contains orbits with two separate groups of orbit angles. In this neighborhood we would expect a corresponding gap in the velocity distribution. Figure 11a shows that we expect gaps in the velocity distribution where there are discontinuities in the spiral structure, and suggests that gaps in the velocity distribution only occur at radii where there are discontinuities in spiral structure and at specific angles where there are spiral arm peaks. We don’t see gaps in every velocity distribution in Figure 8 but do tend to see them where the surface density is high. This is expected as density peaks can occur when multiple pattern peaks coincide.

In the previous section we introduced a first order perturbative orbital model containing a single potential perturbation (equations 11 - 6.1). This model can be extended for multiple patterns giving orbits with radius, radial and tangential velocity as a function of guiding or mean radius to first order in the perturbation strengths

When is small compared to and the orbits are nearly closed as shown in Figure 11a,b.

For a particular neighborhood with the above equations must be inverted to solve for , and as a function of the velocity components. The inversion need not give a unique value of so particles from separated radial regions (in guiding radius ) can simultaneously be present in one local neighborhood. The similarity of terms in imply that there is a close relation between the inverted distribution and the velocity components. This situation is illustrated in Figures 11a,b where transitions between one pattern and another gives neighborhoods containing orbits from different regions of the galaxy and a bifurcated velocity distribution in that neighborhood. Figure 11a shows a transition between one two armed and another two-armed pattern. Figure 11b illustrates a transition between a two-armed and a three-armed pattern. In this figure the inner 6 curves are ellipses however the outer ones are a sum of elliptical and triangle perturbations such as might occur if there are simultaneously two-armed and three-armed spiral density waves. The elliptical and triangular perturbations vary smoothly in orientation angle from curve to curve. The transition region exhibits discontinuities in overlap regions caused by the onset of the three-armed pattern. The blue circle is located at a radius where there is a discontinuity and shows that the angular distribution of orbits is bifurcated. We expect that in such a region there will be gap in the velocity distribution.

A neighborhood with is most likely to contain two separate solutions for if the phases of the arguments for each perturbation have opposite signs. For example if the angle and then and the particles in a neighborhood at will have guiding radii interior to . The orbits will have but greater than the circular velocity at (consistent with the discussion in section 5.5). If the angle then the opposite is true. A neighborhood at can simultaneously contain stars on orbits associated with an interior perturbation near apocenter and stars associated with an outer perturbation near peri-center.

Our spectrograms showed that there are both two-armed and three-armed patterns present in this simulation. The two-armed and three-armed patterns have different radial extents and different pattern speeds. Figure 11b shows that the transition between a two-armed and three-armed pattern can cause a discontinuity in the spiral arm peaks. This type of discontinuity can also give neighborhoods with two sets of orbital orientation angles and so gaps in a velocity distribution, as illustrated by the blue neighborhood in this figure.

We now compare the list of radial discontinuities and gaps with estimated resonance locations identified from the pattern speeds in the simulations. If we only had one strong wave then we would be restricted to a few resonances. However our simulation shows multiple patterns, each with a different radial range. Keeping this in mind, perhaps it is not surprising that the density distribution shows multiple spiral arm discontinuities and the velocity distributions show multiple gaps.

If the spiral patterns are excited at resonances then we expect that they will start or/and end near resonances. In section 4 we discussed likely ranges for these patterns with likely resonances as boundaries. The innermost discontinuity, seen both as gaps in the velocity distribution and in the spiral morphology, lies at radius of about kpc. This lies near the bar’s ILR and is not distant from the ILR of the inner two-armed spiral. The discontinuity at kpc lies near the bar’s corotation resonance, the the ILR of the three-armed spiral and is not distant from the ILR of the inner two-armed spiral and the ILR of the outer two-armed spiral. The discontinuity at a radius is near the OLR of the three-armed spiral and the OLR of the inner two-armed spiral. That at kpc is near the bar’s OLR and the ILR of the outer two-armed spiral. It also may be the outer boundary of the inner two-armed spiral and near its OLR. The division at is not obviously associated with a resonance but could be due to a division between the inner and outer two-armed spirals.

The first order approximation used here has a small divisor problem nearing resonance. To understand the dynamics near resonance a higher order model is required (Contopoulos, 1975). Approaching resonance the epicyclic amplitude increases but does not go to infinity and closed orbits exist on both sides of resonance. Consequently a closed orbit approximation (such as illustrated in Figure 11) may not be invalid even quite close to resonance. As epicyclic amplitudes become high near resonance, Lindblad resonances are particularly likely locations where orbits can cross into nearby neighborhoods. Because the orientation of closed orbits can vary on either side of a Lindblad resonance they can also induce a gap in a local velocity distribution (Dehnen, 2000), also see Figure 4 by Quillen & Minchev (2005). If there are two perturbations, localized regions near resonance can become chaotic (Quillen, 2003) perhaps leading to the truncation of spiral arms. The Hamiltonian model by Quillen (2003) showed that periodic orbits can exist near resonance even when two patterns are present though if there is strong resonance overlap there may be regions lacking nearly circular orbits where localized diffusion can be rapid (Shevchenko, 2011) and heating is possible (Minchev & Quillen, 2006).

The gaps in the velocity distribution and discontinuities in spiral morphology seen in our simulation can be explained with a model containing multiple spiral density waves. The onset radii of each wave introduces the discontinuities in the spiral arm morphology and associated gaps in the velocity distribution. Our simulation shows multiple patterns that are coupled, and there are radii that are likely resonant with more than one pattern. These radii are likely associated with onset radii of waves, gaps in the velocity distribution and discontinuities in the spiral arm morphology.

### 7.1 Clumps like the Hercules Stream

When the neighborhood lies on a discontinuity, two sets of orbits can pass through the neighborhood; see for example the blue neighborhood in Figure 11a. In this case the first set is those associated the spiral arm peak and their velocities would lie on an arc passing through the top right quadrant in the plane. The other set of orbits comes from the inner galaxy and differs in orientation by about 90. This difference in orientation implies that its values will have the opposite sign, and since the orbits have lower guiding radii, they should have lower values. Thus we expect the velocities of stars on the second group of orbits can have low and the opposite sign of and so form a clump on the lower left quadrant in the plane, similar in position to the Hercules stream at km/s in our galaxy (Dehnen & Binney, 1998).

The outer Lindblad resonance with the bar at earlier times in the simulation is at about 9 kpc. If the solar neighborhood lies just outside this resonance with the bar in the Galaxy then the radius in this simulation most matching our position in the Galaxy (just outside the bar’s OLR) is at kpc. We show velocity distributions in neighborhoods with this radius and at different times in Figure 13. At this radius, clumps similar in position to the Hercules stream are seen at angles (with respect to the bar) of 90, 135, 270 and 315 but not at all times. It is rare to see these streams at angles nearly aligned or perpendicular to the bar. This suggests that non-circular motions caused by the bar may be needed to produce the large epicyclic amplitudes corresponding to the strong negative values in the clump.

The orientation of a periodic orbit can depend upon its location. For example, orbits can be aligned with a bar outside the Outer Lindblad resonance and aligned perpendicular to it between this resonance and the corotation resonance. A flips in orbit orientation on either side of a resonance can also cause a gap in the velocity distribution (e.g., Quillen & Minchev 2005). However if the orbital distribution was primarily determined by the bar then we might expect the outer galaxy to look like a ring galaxy. However our simulated galaxy at kpc has a strong two-armed spiral structure. The bar’s outer Lindblad resonance itself may also be a site for coupling between waves. Both the bar and local spiral structure affect the Hercules stream like clumps in the orbital distribution.

The velocity distribution in the solar neighborhood not only displays a stream at km/s (the Hercules stream) but is triangular shaped for velocities with low We show in Figure 14 rightmost panel the velocity distribution of the magnitude-complete, kinematically unbiased sample of 16682 nearby F and G dwarfs by Holmberg et al. (2009). A triangle shaped distribution at lower and a Hercules stream like feature at negative are sometimes seen in our velocity distributions at kpc, near the bar’s outer Lindblad resonance, are both seen for example the rightmost panel of Figure 10. We have looked through our velocity distributions at this and with a bar orientation consistent with the angle (between bar semi-major axis and Sun the Galactic center line) estimated from the SPITZER/GLIMPSE survey (Benjamin et al., 2005) for other examples of velocity distributions that exhibit the gross features present in the solar neighborhood velocity distribution. These are shown in Figure 14 for smaller neighborhoods with size set by along with disk surface density at the same timesteps in Figure 15. In Figure 15 blue circles show the location of neighborhoods used to extract the velocity distributions. The disk has been flipped so that rotation is clockwise rather than counter clockwise and rotated so the extraction neighborhood is oriented at the top of the plot. This has been done so that these images can more readily be compared with models for the Galaxy. The location of the mock Sun is at the blue circle with a galactic longitude of 90 viewing to the right.

At certain times and locations in our simulation the local velocity distribution is similar to that seen in the solar neighborhood at a position consistent with the orientation, length and distance of the Galactic bar. By similar we mean with a triangular shaped feature at low and a Hercules-like stream at low . Weak clumps at low are seen in our velocity distributions, similar to those labelled as low velocity streams such as the Hyades and Pleiades stream in the solar neighborhood (Dehnen & Binney, 1998). We do not match each of the low velocity streams seen in the solar neighborhood velocity distribution but our velocity distributions do exhibit low velocity clumps suggesting that they could be related to spiral waves in the disk. However none the models have have discussed above would easily account for this structure. Our Hercules like stream is more heavily populated by stars than that in the solar neighborhood, probably because our disk is hotter. Our simulated Hercules stream is at lower than that in the solar neighborhood suggesting that a somewhat smaller galactocentric radius would provide a better analog, however at smaller radii it is more difficult to see a strong Hercules like stream in the simulated velocity distributions.

It is interesting to compare the spiral morphologies seen in Figure 15. In all cases the location of the mock Sun is just exterior to a strong spiral arm. This spiral arm would have tangent (as viewed from the mock Sun) at a Galactic longitude of about (to the left on the plot), and approximately consistent with the location of the Centaurus Arm tangent measured by Churchwell et al. (2009) from the GLIMPSE survey counts and denoted as the Crux-Scutum Arm tangent by Vallée (2008). Our simulated galaxy would exhibit only one or two peaks or tangent arms in a projected number count diagram (similar to figure 14 by Churchwell et al. 2009) and so would resemble the observed infrared observations. As our simulation lacks gas we cannot see additional gas rich but star poor spiral arms that are likely present in our Galaxy (Drimmel & Spergel, 2001). While all the images in Figure 15 have a strong spiral arm just interior to the mock Sun, the opposite side of the galaxy has a variety of morphologies. This implies that the solar neighborhood velocity distribution alone is not sufficient to break the degeneracies of a model that includes multiple spiral waves.

## 8 Summary and Discussion

Using an N-body hybrid simulation we have looked for a relationship between structure in local velocity distributions and spiral and bar density waves. Patterns for these waves are identified through Fourier spectrograms. Previous studies (e.g., Rautiainen & Salo 1999), have suggested that there are couplings between waves. The similarity between spectrograms at different stages in our simulations (as the bar slows down) suggests that our simulation also exhibits couplings between waves, with bar and lopsided waves likely driving spiral waves. Even though multiple waves are present, they could be coupled and slowly evolving rather than transient.

We find that arcs in the local velocity distribution are seen in neighborhoods located on arm peaks, whereas interarm regions tend to have lower velocity dispersions. This behavior is understood by considering a series of nested elliptical orbits, appropriate for a cold disk influenced by a single spiral perturbation. Spiral arm peaks are present where ellipses touch or overlap and these locations correspond to neighborhoods containing orbits with a wider range of velocity vector orientations than interarm regions.

Gaps can be seen in the velocity distributions in our simulation all over the simulated galaxy, confirming the early finding by Fux (2000). However, the values of these gaps correspond to a small number of guiding radii that are near resonances with the bar and spiral arm patterns or where there are kinks or discontinuities in the spiral arm morphology. The relation between gaps in the velocity distribution and discontinuities in spiral structure is understood by considering a series of nested ellipses that have a discontinuity in their relative orientation angles. The discontinuity introduces neighborhoods containing overlapping orbits that have a bifurcated distribution of orbit orientations. This bifurcated distribution corresponds to two clumps and a gap separating them in the velocity distribution. Multiple waves, each with a different pattern speed, present in the simulation, introduce radii at which there are discontinuities in the spiral structure. Consequently gaps in the velocity distributions can be seen at multiple radii in the disk.

N-body simulations have the advantage over test particle simulations that the orbits of particles are consistent with the gravitational potential generated by the particles themselves. Additionally N-body simulations naturally contain varying or evolving bar and spiral structures that would require additional and possibly poorly constrained parameters to describe inside test particle simulations. The study here, though preliminary and based on only one hybrid simulation, can be considered to be complimentary to studies carried out with test particle simulations.

### 8.1 Implications for Studies of the Milky Way

From our simulations we have found that interarm regions tend to have lower velocity dispersions than arm peaks. They also tend to lack arcs and gaps in the velocity distribution. The Sun is likely in an interarm region (Vallée, 2008). We may have biased view of the Milky Way disk dynamics because the local velocity distribution does not represent that at other positions in the Galaxy. For example the mean local velocity dispersion in the Galactic disk may have been underestimated because of our interarm location in the Galaxy. Large arcs in the local stellar velocity distribution may be present in nearby spiral arm peaks.

There remains little consensus among different studies for the pattern speed of local spiral structure (e.g. Shaviv 2003; Lepine et al. 2010; Gerhard 2010). The lack of consensus may arise because the presence of more one wave complicates analysis (e.g., Naoz & Shaviv 2007). The Local Spur which is thought to lie between the dominant two-armed or four-armed structures may arise because multiple waves are passing in and out of phase with each other (e.g., Henry et al. 2003; Meidt et al. 2009). As armlets can appear and disappear, a localized short burst of star formation may occur that does not continue to progress around the galaxy as is expected when a sole wave passes through the disk. Evidence for short isolated bursts of star formation in the disk would support interpretation of the Galactic disk in terms of multiple waves simultaneously present in the disk. We have noted that density peaks due to constructive interference of patterns tend to begin at smaller radius and progress to larger radius before fading away. Trails of recently formed clusters with age decreasing with increasing radius may be found in the Galaxy disk.

Recent studies have placed strong constraints on the pattern speed of the Galactic bar from the morphology of the Hercules stream in the local velocity distribution (Dehnen, 2000; Minchev et al., 2007; Gardner et al., 2010). The association of the value of the gap separating the Hercules stream from the rest of the stars with the bar’s outer Lindblad Resonance tightly constrains the bar pattern speed. However features similar to the Hercules stream are seen in our velocity distributions that both associated with the bar’s outer Lindblad resonance and at other locations in the galaxy. These clumps do not remain fixed in time and are affected by the coupled spiral structure in the outer Galaxy. Previous models of the Hercules stream have neglected spiral structure and have been based on test particle simulations, not N-body simulations (with the exception of the study by Fux 2000). If wave coupling is common then the bar’s outer Lindblad resonance may be coincident with a Lindblad resonance with a spiral pattern. Future models could consider both spiral and bar patterns when interpreting the velocity distribution near the bar’s outer Lindblad resonance.

Velocity distributions with morphology similar to that in the solar neighborhood are seen in our simulation at location just exterior to the bar’s outer Lindblad resonance with bar orientation consistent with infrared studies. By similar we mean, showing a Hercules like stream and a triangular shape at lower . The velocity distribution more closely resembles that seen in the solar neighborhood when there is a strong spiral arm, consistent with the observed Centaurus Arm tangent, just interior to the solar neighborhood.

Like many other galaxies, the Milky Way is lopsided (Levine et al., 2006). If coupling between a lopsided motion and a Galactic bar is common then we might expect a strong three armed spiral wave would be present in the solar neighborhood. However, almost all studies have assumed two or four armed spiral structures (Vallée, 2008). If a strong three armed wave is present in the solar neighborhood and its inner Lindblad resonance is coincident with the Galactic bar’s corotation radius (as likely true in our simulation) then we can estimate its pattern speed. At a inner Lindblad resonance the pattern speed is related to the angular rotation rate by . If we set in this relation equal to the bar pattern speed (approximate 1.9 times the angular rotation rate at the solar circle ; Gardner et al. 2010), use a flat rotation curve with and we estimate a pattern speed of putting the Sun near corotation of this three armed pattern. Pattern speeds for spiral structure fall into two groups, a faster set, placing the Sun near the corotation resonance and a slower set placing the Sun near the inner 4:1 Lindblad resonance (see Table 3 by Shaviv 2003). A three armed pattern with pattern speed similar to would have inner Lindblad resonance near the bar’s corotation resonance and so could be coupled to the bar. In our simulation a two spiral pattern is also prominent near the position of the mock solar neighborhood with inner Lindblad resonance near the bar’s corotation rotation resonance. We can estimating the pattern speed of a two armed wave satisfying this condition from the bar’s pattern speed using finding a pattern speed of or about 14 km s kpc consistent with a slower group of pattern speeds estimated a number of studies (see Table 3 by Shaviv 2003).

Lepine et al. (2010) propose that a nearby kink in a spiral arm winding angle is associated with a single Lindblad resonance with a spiral pattern. However our simulations suggest that kinks and discontinuities in the spiral structure not only are associated with Lindblad resonances with a single pattern (as also explored by Quillen & Minchev 2005; Contopoulos 1988) but are also signatures of transitions from one pattern to another. Rather than rare, our simulation suggests that these discontinuities may exist all over the Galaxy.

The mode or wave coupling possibly present in our simulations implies that resonances between different patterns could be commonly coincident. As shown by Minchev & Quillen (2006) this causes chaotic motions in the stellar orbital motions and so accelerated heating (or increasing velocity dispersion) near localized regions of resonance overlap. If waves are coupled in the Milky Way disk then we might except both localized accelerated heating and radial migration (Minchev & Famaey, 2010; Minchev et al., 2011) as a consequence (also see Quillen 2003; Voglis et al. 2006; Shevchenko 2011).

The waves present in our simulation are likely coupled and so not transient. The divisions seen in the velocity distribution we interpret in terms of discontinuities associated with resonances and transitions from one pattern to another rather than a model involving stochastic growth of transient spiral structure (e.g., De Simone et al. 2004).

### 8.2 Future Work

This study only considered one single simulation and it ran only for 1.3 Gyr. The simulation is not ideal as we have undersampled the bulge and halo in order to well resolve the disk, and the energy conservation is not excellent. However, this particular simulation is particularly rich in spiral density waves and so a good one for comparing velocity distributions to structure seen in spectrograms. While our galaxy was simulated in isolation, real galaxies suffer external perturbations. Lopsidedness in our simulation that may be an artifact of our undersampled live halo, is commonly seen in external galaxies (Jog & Combes, 2009). Hence the coupling between bar, lopsided motion and three armed waves that we suspect is present in our simulation may be ubiquitous rather than just a curiosity.

In our simulations we purposely undersampled the halo and bulge particle distributions so that we could better resolve the galaxy disk. However as a result our few and relatively massive halo particles cause spurious heating in the disk. We have measured the radial velocity dispersion in the disk at a radius of 10 kpc and found that it does not increase by more than 10 km/s across the entire simulation. This suggests that heating by massive halo particles is not excessive in this simulation. The similarity between spectrograms in the middle and later part of the simulation suggests that waves are coupled to the bar rather than randomly generated by noise associated with the granularity of our halo during the simulation. Were we to raise the number of massive particles in the disk, halo and bulge, the simulations would exhibit reduced numerical noise and so might have weaker or delayed spiral and bar structures. Future studies should not only consider more accurate simulations with larger numbers of particles but also explore simulations with differently distributed disks, bulges and halos, the role of a cold gaseous component and perturbations due to mergers.

Here we have only qualitatively attempted to relate spiral morphology and patterns to structure seen in the velocity distributions. However future work could aim to bridge the gap between test particle simulations and N-body simulations by understanding the orbits in each feature seen in the velocity distributions. Here we have focused on gross features present seen in the velocity distributions, such as gaps and arcs. Future studies can also aim to understand the finer features as well, that are primarily seen in smaller neighborhoods (e.g., see Figure 10). These studies would help interpret forthcoming pencil beam (e.g., Minchev & Quillen 2008) and large scale velocity and proper motion surveys of stars in the Galactic disk.

Here we have interpreted gaps in the velocity distributions in terms of changes in the shapes of nearly closed orbits associated with different density waves. However future studies could also consider alternate interpretations for structure seen in velocity distributions such as transient spiral arms (De Simone et al., 2004), bifurcations in the families of periodic orbits (Katsanikas et al., 2011), chaotic orbits along spiral arms near a bar (Voglis et al., 2006), phase wrapping following an initial disturbance (Minchev et al., 2009) or individual Lindblad resonances (Quillen & Minchev, 2005).

We thank Larry Widrow for giving us and helping us with his code GalacticICS. We thank NVIDIA for the gift of four Quadro FX 5800 and two GeForce GTX 280 video cards. We thank Evghenii Gaburov and Stefan Harfst for making GRAPE and Sapporo available. Support for this work was provided by NSF through award AST-0907841.

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Pattern | Time | Pattern speed | ILR | ILR | ILR | CR | OLR | OLR | OLR | |
---|---|---|---|---|---|---|---|---|---|---|

(Myr) | (radians/Myr) | (kpc) | (kpc) | (kpc) | (kpc) | (kpc) | (kpc) | (kpc) | (kpc) | |

Bar | 350-950 | 0.040 | 0.5 –6 | 2.4 | 3.6 | 5.1 | 7.2 | 7.6 | 9 | |

Inner two-armed Spiral | 350-950 | 0.030 | 3 –10 | 2.4 | 3.8 | 4.4 | 7.2 | 9.5 | 10 | 11 |

Outer two-armed Spiral | 350-950 | 0.015 | 6 –15 | 4.7 | 7.1 | 9.1 | 13.5 | 17 | 18 | 19 |

Three-armed Spiral | 350-950 | 0.023 | 3 –15 | 4.5 | 9.1 | 12.3 | ||||

Bar | 750-1305 | 0.035 | 0.5 –6 | 3.6 | 3.0 | 6.0 | 8.1 | 9.1 | 10 | |

Inner two-armed spiral | 750-1305 | 0.022 | 3 –10 | 3.2 | 4.9 | 6.0 | 9.5 | 12.0 | 13.3 | 14 |

Outer two-armed Spiral | 750-1305 | 0.013 | 6 –20 | 6.3 | 8.9 | 10.9 | 15 | 19 | 19.5 | 22 |

Three armed Spiral | 750-1305 | 0.023 | 5 –15 | 4.5 | 9.1 | 12.3 |

These pattern speeds are measured from features seen in the spectrograms shown in Figures 4 and 6. The pattern speeds can be approximately converted to units of km s kpc by multiplying by 1000. The times listed in the second column correspond to the range of times used to construct these spectrograms. The radii and are estimated inner and outer radii in kpc of the individual patterns. Here ILR and OLR refer to the inner and outer Lindblad resonances. Likewise ILR and ILR refer to the the and inner Lindblad resonances. When a resonance from one pattern is near that of another pattern, one pattern could be coupled to or driving the other.